THRESHOLD CORRECTIONS IN PRECISION LHC PHYSICS: QED$\otimes$QCD
\footnotesize B.F.L. WARD
With an eye toward LHC processes in which theoretical precisions of 1\% are desired, we introduce the theory of the simultaneous YFS resummation of QED and QCD to compute the size of the expected resummed soft radiative threshold effects in precision studies of heavy particle production at the LHC. Our results show that both QED and QCD soft threshold effects must be controlled to be on the conservative side to achieve such precision goals. \keywords{QCD; QED; Resummation.}
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\begin{document}

\markboth{\footnotesize B.F.L. WARD, C. GLOSSER, S. JADACH, S. A. YOST}
{THRESHOLD CORRECTIONS IN PRECISION LHC PHYSICS: QED$\otimes$QCD}

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\title{THRESHOLD CORRECTIONS IN PRECISION LHC PHYSICS: QED$\otimes$QCD}

\author{\footnotesize B.F.L. WARD}

\address{Department of Physics, Baylor University, One Bear Place \#97316\\
Waco, Texas 76798-7316,
USA
}

\author{\footnotesize C. GLOSSER}

\address{Department of Physics, Southern Illinois University, Box 1654\\
 Edwardsville, IL 62026-1654,  USA
}
\author{\footnotesize S. JADACH}

\address{Institute of Nuclear Physics, ul. Radzikowskiego 152\\
31-342 Krak\'ow, Poland\\
Theory Division, CERN\\
CH-1211 Geneva 23, Switzerland
}

\author{\footnotesize S. A. YOST}

\address{Department of Physics, Baylor University, One Bear Place \#97316\\
 Waco, Texas 76798-7316, USA}



\maketitle

%\pub{Received (Day Month Year)}{Revised (Day Month Year)}

\begin{abstract}
With an eye toward
LHC processes in which theoretical precisions of 1\% are desired, 
we introduce the 
theory of the simultaneous YFS resummation of QED and QCD
to compute the size of the
expected resummed soft radiative threshold effects in 
precision studies of heavy
particle production at the LHC. Our results 
show that both QED and QCD soft threshold effects
must be controlled to be on 
the conservative side to achieve such precision goals.
\keywords{QCD; QED; Resummation.}
\end{abstract}
\section{Introduction}%1
%In high energy collider processes, such as t\=t production at
%FNAL, precision predictions for soft multiple gluon (n(g)) effects
%are already needed: the uncertainty on $m_t$~\cite{mterror}, 
%$\delta m_t = 4.3$ GeV, receives a soft n(g) uncertainty $\sim$ 2-3 GeV,
%for example. 
At the LHC/ILC,
the precision requirements for soft multiple gluon (n(g)) effects
will be even more demanding than at FNAL, where the uncertainty 
on $m_t$~\cite{mterror}, 
$\delta m_t = 4.3$ GeV, receives a soft n(g) uncertainty $\sim$ 2-3 GeV, 
and soft n(g) MC exponentiation results will be an 
important part of the necessary theory -- YFS exponentiated
${\cal O}(\alpha_s^2)L$ calculations, {\em in the presence of parton showers}, 
on an event-by-event basis.

%How relevant are QED higher order corrections when QCD is
%controlled at $\sim 1\%$ precision? 
As many authors~\cite{qcdlit} 
prepare the
necessary results that would lead to such a precision on QCD
for LHC processes, the QED corrections need to be addressed as well.
Estimates by Refs.~\cite{cern2000,spies,james1,roth,james2}
show that one gets few per mille
effects from QED corrections to structure function evolution.
In this paper, estimate the size of QED corrections at threshold
%The well-known possible enhancement of QED corrections at threshold,
%especially in resonance production, leads us to estimate
%how big are these effects 
at the LHC.

Treating simultaneously QED and QCD in the respective 
YFS~\cite{yfs,yfs1} exponentiation we discuss
%to estimate the role of the QED 
threshold effects at the LHC 
in the candidate luminometry~\cite{lhclum,fnallum} processes
$pp\rightarrow V +n(\gamma)+m(g)+X\rightarrow \bar{\ell} \ell'
+n'(\gamma)+m(g)+X$, where 
$V=W^\pm,Z$,and $\ell = e,\mu,~\ell'=\nu_e,\nu_\mu ( e,\mu )$
respectively for $V=W^+ ( Z )$, and  
$\ell = \nu_e,\nu_\mu,~\ell'= e,\mu$ respectively for $V = W^-$.
%Precision studies of these processes have been proposed for
%luminometry at the LHC~\cite{lhclum} and at FNAL~\cite{fnallum}, 
%where 2-3\% is the target
%precision tag for the LHC, for example. The latter would indeed require
%a theoretical precision tag of $\sim 1\%$ in order that
%the theory error not figure too prominently in the over-all 
%precision. 

%Our discussion is organized as follows. After giving a brief
%review of the YFS theory and its extension to QCD in Sect. 2,
%in Sect. 3 we introduce QED$\otimes$QCD YFS exponentiation.
%In Sect. 4, we apply the new development to the 
%threshold corrections in single
%V production at the LHC and at FNAL.\par
% Section 5 contains some
%concluding remarks.\par

\section{YFS Theory and its Extension to QCD}

In Refs.~\cite{yfs1} the renormalization group improved YFS theory~\cite{bflw1yfs} 
for $e^+(p_1)e^-(q_1)\rightarrow \bar{f}(p_2) f(q_2) +n(\gamma)(k_1,\cdot,k_n)$
is realized by Monte Carlo methods, where the respective cross section
$d\sigma_{exp}$ and all of the attendant
IR functions, $\{B,\tilde B, D, \tilde S\}$, and hard photon residuals
,$\{\bar\beta_n(k_1,\ldots,k_n)\}$, are specified in Refs.~\cite{yfs1}.
%renormalization group improved YFS theory~\cite{bflw1yfs} gives,{\small
%\begin{equation}
%d\sigma_{exp}=e^{2\alpha\,Re\,B+2\alpha\,
%\tilde B}\sum_{n=0}^\infty{1\over n!}\int\prod_{j=1}^n{d^3k_j\over k_j^0
%}\int {d^4y\over(2\pi)^4}e^{iy(p_1+q_1-p_2-q_2-\sum_jk_j)+D}\cr
%\qquad\bar\beta_n(k_1,\dots,k_n){d^3p_2d^3q_2\over p_2^0q_2^0}
%\label{eqone1}\end{equation}}\noindent
%where the YFS infrared functions $\tilde B,~B$ and $D$ are
%known. 
%and {\small
%\begin{eqnarray}
%2\alpha\,\tilde B &= \int^{k\le K_{max}}{d^3k\over k_0}\tilde S(k)\nonumber\\
%                  D &=\int d^3k{\tilde S(k)\over k^0}\left(e^{-iy\cdot k}-\theta(K_{max}-k)\right)
%\label{eqtwo}
%\end{eqnarray} }
%for the infrared emission factor{\small
%\begin{equation}
%\tilde S(k)= {\alpha\over4\pi^2}[Q_fQ_{
%{\llap{\phantom f}^{\sstl(}\bar f^{\sstl^)}{}}'}\left({p_1\over p_1\cdot k}-{q_1
%\over q_1\cdot k}\right)^2+(\dots)]
%\label{eqthree}
%\end{equation} }
%if $Q_f$ is the electric charge of $f$ in units of the positron charge. 
%For example, the YFS hard photon residuals $\bar\beta_i$, 
%$i=0,1,2$,
%are given in the first paper in Ref.~\cite{yfs1} and realize the
%YFS exponentiated exact ${\cal O}(\alpha)$
%and LL ${\cal O}(\alpha^2)$
%cross section for Bhabha scattering via a
%corresponding Monte Carlo realization of the YFS theory.
In Refs.~\cite{qcdref,dglapsyn}
%%hep-ph/0210357(ICHEP02), Acta Phys.Polon.B33,1543-1558,2002,
we have extended the YFS theory to QCD:
the net result is that in the analogous YFS theory we have the 
replacements $2\alpha\,Re\,B+2\alpha\,\tilde B\rightarrow SUM_{IR}(QCD)$,
$D\rightarrow D_\rQCD$, and $\bar\beta_n(k_1,\dots,k_n)\rightarrow \tilde{\bar\beta}_n(k_1,\ldots,k_n)$, where the QCD YFS functions are defined
in Ref.~\cite{qcdref} and the
%{\small
%\begin{equation}
%\begin{split}
%d\hat\sigma_{\rm exp}&= \sum_n d\hat\sigma^n 
%         =e^{\rm SUM_{IR}(QCD)}\sum_{n=0}^\infty\int\prod_{j=1}^n{d^3
%k_j\over k_j}\\&\quad\int{d^4y\over(2\pi)^4}e^{iy\cdot(P_1+P_2-Q_1-Q_2-\sum k_j)+
%D_\rQCD}\\
%&*\tilde{\bar\beta}_n(k_1,\ldots,k_n){d^3P_2\over P_2^{\,0}}{d^3Q_2\over
%Q_2^{\,0}}
%\end{split}
%\label{subp15a}
%\end{equation}}\noindent
gluon residuals~\cite{qcdref} 
$\tilde{\bar\beta}_n(k_1,\ldots,k_n)$
%, defined by Ref.~\cite{qcdref}, 
%\[\tilde{\bar\beta}_n(k_1,\ldots,k_n)= \sum_{\ell=0}^\infty 
%\tilde{\bar\beta}^{(\ell)}_n(k_1,\ldots,k_n)\]
%,where is the number of virtual gluon loops, 
are free of all infrared divergences to all 
orders in $\alpha_s(Q)$. 
%The functions
%$SUM_{IR}(QCD), D_\rQCD$, together with 
%the basic infrared functions 
%$B^{nls}_{QCD},{\tilde B}^{nls}_{QCD},{\tilde S}^{nls}_{QCD}$ 
%are specified in Ref.~\cite{qcdref}.
The genuine non-Abelian IR physics is encoded~\cite{qcdref} here
in the $\tilde{\bar\beta}_j$.
%We call attention to the essential
%compensation between
%the left over genuine non-Abelian IR virtual and real singularities
%between $\int dPh\bar\beta_n$ and $\int dPh\bar\beta_{n+1}$ respectively
%that really allows us to isolate $\tilde{\bar\beta}_j$ and distinguishes
%QCD from QED, where no such compensation occurs.
The YFS resummation which we discuss here is
fully consistent with that of Refs.~\cite{ster,cattrent}.
See Ref.~\cite{CG1} for more discussion of this point.

\section{Extension to QED$\otimes$QCD and QCED}

Simultaneous exponentiation of
QED and QCD higher order effects
gives~\cite{CG1}
%{\small
%\begin{eqnarray}
%B^{nls}_{QCD} \rightarrow B^{nls}_{QCD}+B^{nls}_{QED}\equiv B^{nls}_{QCED},\cr
%{\tilde B}^{nls}_{QCD}\rightarrow {\tilde B}^{nls}_{QCD}+{\tilde B}^{nls}_{QED}\equiv {\tilde B}^{nls}_{QCED}, \cr
%{\tilde S}^{nls}_{QCD}\rightarrow {\tilde S}^{nls}_{QCD}+{\tilde S}^{nls}_{QED}\equiv {\tilde S}^{nls}_{QCED}
%\label{irsub}
%\end{eqnarray}} 
%which leads to 
{\small
\begin{equation}
\begin{split}
d\hat\sigma_{\rm exp} &= e^{\rm SUM_{IR}(QCED)}
   \sum_{{m,n}=0}^\infty\int\prod_{j_1=1}^m\frac{d^3k_{j_1}}{k_{j_1}} 
\prod_{j_2=1}^n\frac{d^3{k'}_{j_2}}{{k'}_{j_2}}
\int\frac{d^4y}{(2\pi)^4}\\&e^{iy\cdot(p_1+q_1-p_2-q_2-\sum k_{j_1}-\sum {k'}_{j_2})+
D_\rQCED} 
\tilde{\bar\beta}_{m,n}(k_1,\ldots,k_m;k'_1,\ldots,k'_n)\frac{d^3p_2}{p_2^{\,0}}\frac{d^3q_2}{q_2^{\,0}},
\end{split}
\label{subp15b}
\end{equation}}\noindent
where the new YFS functions, $SUM_{IR}(QCED),~D_\rQCED$ and
$\tilde{\bar\beta}_{m,n}(k_1,\ldots,k_m;k'_1,\ldots,k'_n)$, where the
latter has $m$ hard gluons and $n$ hard photons, are
defined in Ref.~\cite{CG1}.
% the residuals
%$\tilde{\bar\beta}_{m,n}(k_1,\ldots,k_m;k'_1,\ldots,k'_n)$, with $m$ hard gluons and $n$ hard photons,
%represent the successive application
%of the YFS expansion first for QCD and subsequently for QED. 
%The
%functions ${\rm SUM_{IR}(QCED)},D_\rQCED$ are determined
%from their analoga ${\rm SUM_{IR}(QCD)},D_\rQCD$ via the
%substitutions in (\ref{irsub}) everywhere in expressions for the
%latter functions given in Refs.~\cite{qcdref}. 
%{\small 
%\begin{eqnarray}
%{\rm SUM_{IR}(QCED)}=2\alpha_s\Re B^{nls}_{QCED}+2\alpha_s{\tilde B}^{nls}_{QCED}\cr
%D_\rQCED=\int \frac{dk}{k^0}\left(e^{-iky}-\theta(K_{max}-k^0)\right){\tilde S}^{nls}_{QCED}
%\label{irfns}
%\end{eqnarray}}
%where $K_{max}$ is a dummy parameter -- here the same for QCD and QED.
The infrared algebra QCED~\cite{CG1} obtains: the average Bjorken $x$ values
for the QED and QCD emissions imply~\cite{CG1}
%
%\begin{eqnarray}
%$x_{avg}(QED)\cong \gamma(QED)/(1+\gamma(QED))$ and
%$x_{avg}(QCD)\cong \gamma(QCD)/(1+\gamma(QCD))$
%\nonumber
%\end{eqnarray}
%where
%$\gamma(A)=\frac{2\alpha_{A}{\cal C}_A}{\pi}(L_s
%-1)$, $A=QED,QCD$, with
%${\cal C}_A=Q_f^2, C_F$, respectively, for 
%$A=QED,QCD$ and the big log $L_s$, imply that
that QCD dominant corrections happen an
order of magnitude earlier than those for QED so
that the leading $\tilde{\bar\beta}_{0,0}^{(0,0)}$-level
gives a good estimate of the size of the effects we study.

\section{QED$\otimes$QCD Threshold Corrections at the LHC} 

%We shall apply the new simultaneous QED$\otimes$QCD exponentiation
%calculus to the single Z production with leptonic decay 
%at the LHC ( and at FNAL)
%to focus on the ISR alone, for definiteness.
%See also the work of Refs.~\cite{baurall,ditt,russ} for exact ${\cal O}(\alpha)$ results and Refs.~\cite{van1,van2,anas}
%for exact ${\cal O}(\alpha_s^2)$ results.

For the basic formula (we use the standard notation here~\cite{CG1}){\small
\begin{equation}
d\sigma_{exp}(pp\rightarrow V+X\rightarrow \bar\ell \ell'+X') =
\sum_{i,j}\int dx_idx_j F_i(x_i)F_j(x_j)d\hat\sigma_{exp}(x_ix_js),
\label{sigtot} 
\end{equation}}\noindent
we use the result in (\ref{subp15b}) for $V=Z$ here with semi-analytical
methods and structure functions from Ref.~\cite{mrst1}. See also the work of Refs.~\cite{baurall,ditt,russ,van1,van2,anas}.
A Monte Carlo realization will appear elsewhere~\cite{elsewh},
wherein we will ultimately use HERWIG~\cite{herwig}, 
PYTHIA~\cite{pythia} and /or the new shower algorithm in Ref.~\cite{jadskrz} 
in lieu of the $\{F_i\}$ and thereby, in principle, improve on the shower/exact
result combination in Ref.~\cite{frixw}. Due to its lack of the appropriate color coherence~\cite{mmm}, we do not consider ISAJET~\cite{isajet} here.

%We {\bf do not} attempt to replace HERWIG~\cite{herwig} 
%and/or PYTHIA~\cite{pythia} --
%we intend to combine our exact YFS calculus with HERWIG and/or PYTHIA
%{\bf by using the latter in lieu of the $\{F_i\}$}.
%This combination of theoretical constructs can be 
%systematically improved with
%exact results order-by-order in $\alpha_s$, where  
%currently the state of the art in such
%a calculation is the work of Frixione and Webber in Ref.~\cite{frixw}
%which accomplishes the combination of an exact ${\cal O}(\alpha_s)$
%correction with HERWIG. We note that, even in this latter result,
%the gluon azimuthal angle is averaged in the combination. We note that
%the recent alternative parton shower algorithm by Jadach and 
%Skrzypek in Ref.~\cite{jadskrz} can also be used in our theoretical
%construction here. Due to its lack of the appropriate color coherence~\cite{mmm}, we do not consider ISAJET~\cite{isajet} here.

We compute , with and without QED, the ratio
$r_{exp}=\sigma_{exp}/\sigma_{Born}$
to get the results
(We stress that we {\em do not} use the narrow resonance approximation here.){\small
\begin{equation}
r_{exp}=
\begin{cases}
1.1901&, \text{QCED}\equiv \text{QCD+QED,~~LHC}\\
1.1872&, \text{QCD,~~LHC}\\
1.1911&, \text{QCED}\equiv \text{QCD+QED,~~Tevatron}\\
1.1879&, \text{QCD,~~Tevatron.}\\
\end{cases}
\label{res1}
\end{equation}}\noindent
QED is at the level of .3\% at both LHC and FNAL
\footnote{This is stable under scale variations~\cite{CG1}.}.
We agree with the results in Refs.~\cite{baurall,ditt,russ,van1,van2}.
%on both of the respective sizes of the QED and QCD effects.
The QED effect is similar in size to structure function
results in Refs.~\cite{cern2000,spies,james1,roth,james2}.

%\section{Conclusions}

%YFS theory (EEX and CEEX) extends to 
%non-Abelian gauge theory and allows simultaneous exponentiation of 
%QED and QCD. For QED$\otimes$QCD we find that
%full MC event generator realization is possible in a way that
%combines our calculus with Herwig and Pythia in principle.
%Semi-analytical results for QED (and QCD) threshold effects agree 
%with literature on Z production. As QED is at the .3\% level, 
%it is needed for 1\% LHC theory predictions.
%A firm basis for the complete ${\cal O}(\alpha_s^2,\alpha\alpha_s,\alpha^2)$
%results needed for the FNAL/LHC/RHIC/TESLA/ILC physics 
%has been demonstrated and all of the latter are in progress.

\section*{Acknowledgments}
Two of us ( S.J. and B.F.L.W.)
thank Profs. S. Bethke and L. Stodolsky for the support and kind
hospitality of the MPI, Munich, while a part of this work was
completed. This work was supported partly by US DoE contract
DE-FG05-91ER40627 and by NATO grants PST.CLG.97751,980342.

\begin{thebibliography}{0}
\bibitem{mterror} J. G. da Costa, in {\it Proc. ICHEP04}, in press; D. Denisov,
, {\it ibid.}.
\bibitem{qcdlit} See for example T. Gehrmann, in {\it Proc. LP2003},
ed. C.Sazama ( FNAL, 2004 ), in press; J. Stirling, in {\it Proc. ICHEP04},
in press, and references therein.
\bibitem{cern2000} S. Haywood, P.R. Hobson, W. Hollik and Z. Kunszt,
in {\it Proc. 1999 CERN Workshop on Standard Model Physics ( and more )
at the LHC, CERN-2000-004}, eds. G. Altarelli and M.L. Mangano,( CERN,
Geneva, 2000 ) p. 122.
\bibitem{spies} H. Spiesberger,{\it Phys. Rev.} D{\bf 52}, 4936 (1995).
\bibitem{james1} W.J. Stirling,''Electroweak Effects in Parton Distribution 
Functions'', talk presented at ESF Exploratory Workshop,
{\it Electroweak Radiative Corrections to Hadronic Observables at TeV
Energies }, Durham, Sept., 2003.
\bibitem{roth} M. Roth and S. Weinzierl,{\it Phys. Lett.} {\bf B590}, 190 (2004).
\bibitem{james2} J. Stirling, in {\it Proc. ICHEP04}, in press.
\bibitem{yfs}D.~R.~Yennie, S.~C.~Frautschi, and H.~Suura,{\it Ann. Phys.} {\bf 13}, 379 (1961);
see also K.~T.~Mahanthappa, {\it Phys.~Rev.}~{\bf 126}, 329 (1962), for a related analysis.
\bibitem{yfs1} See also S. Jadach {\it et al.},{\it Comput. Phys. Commun.} {\bf 102}, 229 (1997); S. Jadach, M. Skrzypek and B.F.L. Ward,{\it Phys.Rev.} D{\bf 55}, 1206 (1997); S. Jadach, B.F.L. Ward and Z. Was,{\it Phys. Rev.} D{\bf 63}, 113009 (2001); S. Jadach, B.F.L. Ward and Z. Was, {\it Comp. Phys. Commun.} {\bf 130}, 260 (2000); S. Jadach {\it et al.}, {\it ibid.}{\bf 140}, 432, 475 (2001).
\bibitem{lhclum} M. Dittmar, F. Pauss and D. Zurcher,{\it Phys. Rev.} D{\bf 56}, 7284 (1997); M. Rijssenbeek, in {\it Proc. HCP2002}, ed. M. Erdmann,( 
Karlsruhe, 2002 )p. 424; M. Dittmar, {\it ibid.},p.431.
\bibitem{fnallum} S. Klimenko, in {\it Proc. HCP2002}, ed. M. Erdmann,( 
Karlsruhe, 2002 ) p.413.
%\bibitem{thorne1} R. Thorne, private communication, 2003. 
%\bibitem{yfs}D.~R.~Yennie, S.~C.~Frautschi, and H.~Suura,{\it Ann. Phys.} {\bf 13} (1961) 379;\newline
%see also K.~T.~Mahanthappa, {\it Phys.~Rev.}~{\bf 126} (1962) 329, for a related analysis.
%\bibitem{yfs1} See also S. Jadach {\it et al.},{\it Comput. Phys. Commun.} {\bf 102}
%(1997) 229; S. Jadach, M. Skrzypek and B.F.L. Ward,{\it Phys.Rev.} D{\bf 55} (1997) 1206; S. Jadach, B.F.L. Ward and Z. Was,{\it Phys. Rev.} D{\bf 63} (2001) 113009;
%S. Jadach, B.F.L. Ward and Z. Was, {\it Comp. Phys. Commun.} {\bf 130} (2000) 260;
%S. Jadach {\it et al.}, {\it ibid.}{\bf 140} (2001) 432, 475,
%and references therein.
\bibitem{bflw1yfs}B.F.L. Ward,{\it Phys. Rev.} D{\bf 36}, 939 (1987).
\bibitem{qcdref} B.F.L. Ward and S. Jadach, {\it Acta Phys.Polon.} {\bf B33}, 1543 (2002); in {\it Proc. ICHEP2002},
ed. S. Bentvelsen {\it et al.},( North Holland, Amsterdam, 2003 ) p. 275.
\bibitem{dglapsyn} B.F.L. Ward and S. Jadach,{\it Mod. Phys. Lett.} A {\bf 14},
 491 (1999).
\bibitem{ster}G. Sterman,{\it Nucl. Phys.}B {\bf 281}, 310 (1987).
\bibitem{cattrent} S. Catani and L. Trentadue,
{\it Nucl. Phys.}B {\bf 327}, 323 (1989); {\it ibid.} {\bf 353}, 183 (1991). 
\bibitem{CG1} C. Glosser {\it et al.},{\it Mod. Phys. Lett.} A {\bf 19}, 2119 (2004).
%\bibitem{top1} S. Jadach {\it et al.}, Mod. Phys. Lett. {\bf A12} (1997)
%2425.
%\bibitem{dglapsyn} B.F.L. Ward and S. Jadach, Mod. Phys. Lett. {\bf A14}
%(1999) 491.
%\bibitem{catani}S. Catani, M.L. Mangano, P. Nason and L. Trentadue,
%Phys. Lett. {\bf B378} (1996) 329.
%\bibitem{dglapsyn} B.F.L. Ward and S. Jadach, Mod. Phys. Lett. {\bf A14}
%(1999) 491.
%\bibitem{ster}G. Sterman, Nucl. Phys. {\bf B281} (1987) 310.
%\bibitem{cattrent} S. Catani and L. Trentadue, 
%Nucl. Phys.{\bf B327} (1989) 323; {\it ibid.} {\bf B353} (1991) 183.
\bibitem{baurall}  U. Baur {\it et al.}, 
{\it Phys. Rev.} D {\bf 57}, 199 (1998); {\it ibid.}{\bf 59}, 013002 (1998); {\it ibid.}{\bf 65}, 033007 (2002), and references therein.
\bibitem{ditt} S. Dittmaier and M. Kramer, {\it Phys. Rev.} D{\bf 65}, 073007 (2002), and references therein.
\bibitem{russ} Z. A. Zykunov,{\it Eur. Phys. J.}C{\bf 3}, 9 (2001), and references therein.  
\bibitem{van1}R. Hamberg, W. L. van Neerven and T Matsuura, 
{\it Nucl. Phys.}B{\bf 359}, 343 (1991).
\bibitem{van2}W.L. van Neerven and E.B. Zijlstra, {\it Nucl. Phys.} B{\bf 382} (1992)
11; {\it ibid.}{\bf 680}, 513 (2004).
\bibitem{anas} C. Anastasiou {\it et al.},{\it Phys. Rev.} D{\bf 69},094008 (2004).
\bibitem{mrst1} A.D. Martin \underline{et al.}, {\it Phys. Rev.} D{\bf 51}, 4756 (1995).
\bibitem{elsewh}
S. Jadach {\it et al.}, to appear.
%\bibitem{app1}B.F.L. Ward and S. Jadach,
%Acta Phys. Polon. {\bf B33} (2002) 1543, and references therein.
%\bibitem{ichep02}B.F.L. Ward and S. Jadach, 
%``QCD Soft Gluon Exponentiation: YFS MC Approach'', in {\it Proc. ICHEP2002},
%ed. S. Bentvelsen {\it et al.},( North Holland, Amsterdam, 2003 ) p. 275
%and references therein.
\bibitem{herwig} G. Corcella et al., hep-ph/0210213, and refrences therein.
\bibitem{pythia} T. Sjostrand et al., hep-ph/0308153.
%\bibitem{isajet} F. Paige et al., hep-ph/0312045, and references therein.
\bibitem{frixw} S. Frixione and B. Webber, hep-ph/0402116, and references 
therein.
\bibitem{jadskrz} S. Jadach and M. Skrzypek, {\it Acta Phys. 
Pol.} {\bf B35}, 735 (2004). 
\bibitem{mmm} M. Michelangelo, private communication, 2004.
\bibitem{isajet} F. Paige et al., hep-ph/0312045, and references therein.
%S. Jadach {\it et al.}, to appear.
%\bibitem{baur0} U. Baur, S. Keller and W.K. Sakumoto, 
%Phys. Rev. {\bf D57} (1998) 199.
%\bibitem{baur} U. Baur, S. Keller and D. Wackeroth, Phys. Rev. D{\bf 59} (1998) 013002, and references therein.
%\bibitem{hol} U. Baur {\it et al.}, Phys. Rev. D{\bf 65} (2002) 033007, and references therein.
%\bibitem{ditt} S. Dittmaier and M. Kramer, Phys. Rev. D{\bf 65} (2002) 073007, and references therein
%\bibitem{russ} Z. A. Zykunov, Eur. Phys. J. {\bf C3} (2001) 9, and references therein.
%\bibitem{born1}
%S. Jadach, B.F.L. Ward and Z. Was, Phys. Rev. D{\bf 63} (2001) 113009.
%\bibitem{van1}R. Hamberg, W. L. van Neerven and T Matsuura, 
%Nucl. Phys. {\bf B359} (1991) 343.
%\bibitem{van2}W.L. van Neerven and E.B. Zijlstra, Nucl.Phys. {\bf B382} (1992)
%11, and references therein.
%\bibitem{mrst1} A.D. Martin {\it et al.}, Phys. Rev. D{\bf 51} (1995) 4756.

%\bibitem{ja}
%C Jarlskog in {\it CP Violation}, ed. C Jarlskog
%(World Scientific, Singapore, 1988).

%\bibitem{ma}
%L. Maiani, {\it Phys. Lett.} B {\bf 62}, 183 (1976).

%\bibitem{bu}
%J.D. Bjorken and I. Dunietz, {\it Phys. Rev.} D {\bf 36}, 2109 (1987).

%\bibitem{Marnelius} R. Marnelius, {\it Acta Phys. Pol.} 
%{\bf B13}, 669 (1982).

%\bibitem{Bjorken} J. D. Bjorken, in {\it Lecture Notes on Current-Induced 
%Reactions}, ed. J. Komer {\it et al.} (Springer, Heidelberg, 1975).

%\bibitem{Bohr} A. Bohr and B. R. Mottelson, {\it Nuclear Structure} 
%(Benjamin, New York, 1969), Vol.~1, pp.~100--102.

%\bibitem{Webb} R. C. Webb, PhD thesis, Princeton University, 1972.

%\bibitem{Toimela} T. Toimela, Helsinki Research Institute for 
%Theoretical Physics, Report No. HU-TFT-82-37, 1982 (unpublished).
\end{thebibliography}

\end{document}



hep-ph0411047.extracted_bib
\begin{thebibliography}{0}
\bibitem{mterror} J. G. da Costa, in {\it Proc. ICHEP04}, in press; D. Denisov,
, {\it ibid.}.
\bibitem{qcdlit} See for example T. Gehrmann, in {\it Proc. LP2003},
ed. C.Sazama ( FNAL, 2004 ), in press; J. Stirling, in {\it Proc. ICHEP04},
in press, and references therein.
\bibitem{cern2000} S. Haywood, P.R. Hobson, W. Hollik and Z. Kunszt,
in {\it Proc. 1999 CERN Workshop on Standard Model Physics ( and more )
at the LHC, CERN-2000-004}, eds. G. Altarelli and M.L. Mangano,( CERN,
Geneva, 2000 ) p. 122.
\bibitem{spies} H. Spiesberger,{\it Phys. Rev.} D{\bf 52}, 4936 (1995).
\bibitem{james1} W.J. Stirling,''Electroweak Effects in Parton Distribution 
Functions'', talk presented at ESF Exploratory Workshop,
{\it Electroweak Radiative Corrections to Hadronic Observables at TeV
Energies }, Durham, Sept., 2003.
\bibitem{roth} M. Roth and S. Weinzierl,{\it Phys. Lett.} {\bf B590}, 190 (2004).
\bibitem{james2} J. Stirling, in {\it Proc. ICHEP04}, in press.
\bibitem{yfs}D.~R.~Yennie, S.~C.~Frautschi, and H.~Suura,{\it Ann. Phys.} {\bf 13}, 379 (1961);
see also K.~T.~Mahanthappa, {\it Phys.~Rev.}~{\bf 126}, 329 (1962), for a related analysis.
\bibitem{yfs1} See also S. Jadach {\it et al.},{\it Comput. Phys. Commun.} {\bf 102}, 229 (1997); S. Jadach, M. Skrzypek and B.F.L. Ward,{\it Phys.Rev.} D{\bf 55}, 1206 (1997); S. Jadach, B.F.L. Ward and Z. Was,{\it Phys. Rev.} D{\bf 63}, 113009 (2001); S. Jadach, B.F.L. Ward and Z. Was, {\it Comp. Phys. Commun.} {\bf 130}, 260 (2000); S. Jadach {\it et al.}, {\it ibid.}{\bf 140}, 432, 475 (2001).
\bibitem{lhclum} M. Dittmar, F. Pauss and D. Zurcher,{\it Phys. Rev.} D{\bf 56}, 7284 (1997); M. Rijssenbeek, in {\it Proc. HCP2002}, ed. M. Erdmann,( 
Karlsruhe, 2002 )p. 424; M. Dittmar, {\it ibid.},p.431.
\bibitem{fnallum} S. Klimenko, in {\it Proc. HCP2002}, ed. M. Erdmann,( 
Karlsruhe, 2002 ) p.413.
%\bibitem{thorne1} R. Thorne, private communication, 2003. 
%\bibitem{yfs}D.~R.~Yennie, S.~C.~Frautschi, and H.~Suura,{\it Ann. Phys.} {\bf 13} (1961) 379;\newline
%see also K.~T.~Mahanthappa, {\it Phys.~Rev.}~{\bf 126} (1962) 329, for a related analysis.
%\bibitem{yfs1} See also S. Jadach {\it et al.},{\it Comput. Phys. Commun.} {\bf 102}
%(1997) 229; S. Jadach, M. Skrzypek and B.F.L. Ward,{\it Phys.Rev.} D{\bf 55} (1997) 1206; S. Jadach, B.F.L. Ward and Z. Was,{\it Phys. Rev.} D{\bf 63} (2001) 113009;
%S. Jadach, B.F.L. Ward and Z. Was, {\it Comp. Phys. Commun.} {\bf 130} (2000) 260;
%S. Jadach {\it et al.}, {\it ibid.}{\bf 140} (2001) 432, 475,
%and references therein.
\bibitem{bflw1yfs}B.F.L. Ward,{\it Phys. Rev.} D{\bf 36}, 939 (1987).
\bibitem{qcdref} B.F.L. Ward and S. Jadach, {\it Acta Phys.Polon.} {\bf B33}, 1543 (2002); in {\it Proc. ICHEP2002},
ed. S. Bentvelsen {\it et al.},( North Holland, Amsterdam, 2003 ) p. 275.
\bibitem{dglapsyn} B.F.L. Ward and S. Jadach,{\it Mod. Phys. Lett.} A {\bf 14},
 491 (1999).
\bibitem{ster}G. Sterman,{\it Nucl. Phys.}B {\bf 281}, 310 (1987).
\bibitem{cattrent} S. Catani and L. Trentadue,
{\it Nucl. Phys.}B {\bf 327}, 323 (1989); {\it ibid.} {\bf 353}, 183 (1991). 
\bibitem{CG1} C. Glosser {\it et al.},{\it Mod. Phys. Lett.} A {\bf 19}, 2119 (2004).
%\bibitem{top1} S. Jadach {\it et al.}, Mod. Phys. Lett. {\bf A12} (1997)
%2425.
%\bibitem{dglapsyn} B.F.L. Ward and S. Jadach, Mod. Phys. Lett. {\bf A14}
%(1999) 491.
%\bibitem{catani}S. Catani, M.L. Mangano, P. Nason and L. Trentadue,
%Phys. Lett. {\bf B378} (1996) 329.
%\bibitem{dglapsyn} B.F.L. Ward and S. Jadach, Mod. Phys. Lett. {\bf A14}
%(1999) 491.
%\bibitem{ster}G. Sterman, Nucl. Phys. {\bf B281} (1987) 310.
%\bibitem{cattrent} S. Catani and L. Trentadue, 
%Nucl. Phys.{\bf B327} (1989) 323; {\it ibid.} {\bf B353} (1991) 183.
\bibitem{baurall}  U. Baur {\it et al.}, 
{\it Phys. Rev.} D {\bf 57}, 199 (1998); {\it ibid.}{\bf 59}, 013002 (1998); {\it ibid.}{\bf 65}, 033007 (2002), and references therein.
\bibitem{ditt} S. Dittmaier and M. Kramer, {\it Phys. Rev.} D{\bf 65}, 073007 (2002), and references therein.
\bibitem{russ} Z. A. Zykunov,{\it Eur. Phys. J.}C{\bf 3}, 9 (2001), and references therein.  
\bibitem{van1}R. Hamberg, W. L. van Neerven and T Matsuura, 
{\it Nucl. Phys.}B{\bf 359}, 343 (1991).
\bibitem{van2}W.L. van Neerven and E.B. Zijlstra, {\it Nucl. Phys.} B{\bf 382} (1992)
11; {\it ibid.}{\bf 680}, 513 (2004).
\bibitem{anas} C. Anastasiou {\it et al.},{\it Phys. Rev.} D{\bf 69},094008 (2004).
\bibitem{mrst1} A.D. Martin \underline{et al.}, {\it Phys. Rev.} D{\bf 51}, 4756 (1995).
\bibitem{elsewh}
S. Jadach {\it et al.}, to appear.
%\bibitem{app1}B.F.L. Ward and S. Jadach,
%Acta Phys. Polon. {\bf B33} (2002) 1543, and references therein.
%\bibitem{ichep02}B.F.L. Ward and S. Jadach, 
%``QCD Soft Gluon Exponentiation: YFS MC Approach'', in {\it Proc. ICHEP2002},
%ed. S. Bentvelsen {\it et al.},( North Holland, Amsterdam, 2003 ) p. 275
%and references therein.
\bibitem{herwig} G. Corcella et al., hep-ph/0210213, and refrences therein.
\bibitem{pythia} T. Sjostrand et al., hep-ph/0308153.
%\bibitem{isajet} F. Paige et al., hep-ph/0312045, and references therein.
\bibitem{frixw} S. Frixione and B. Webber, hep-ph/0402116, and references 
therein.
\bibitem{jadskrz} S. Jadach and M. Skrzypek, {\it Acta Phys. 
Pol.} {\bf B35}, 735 (2004). 
\bibitem{mmm} M. Michelangelo, private communication, 2004.
\bibitem{isajet} F. Paige et al., hep-ph/0312045, and references therein.
%S. Jadach {\it et al.}, to appear.
%\bibitem{baur0} U. Baur, S. Keller and W.K. Sakumoto, 
%Phys. Rev. {\bf D57} (1998) 199.
%\bibitem{baur} U. Baur, S. Keller and D. Wackeroth, Phys. Rev. D{\bf 59} (1998) 013002, and references therein.
%\bibitem{hol} U. Baur {\it et al.}, Phys. Rev. D{\bf 65} (2002) 033007, and references therein.
%\bibitem{ditt} S. Dittmaier and M. Kramer, Phys. Rev. D{\bf 65} (2002) 073007, and references therein
%\bibitem{russ} Z. A. Zykunov, Eur. Phys. J. {\bf C3} (2001) 9, and references therein.
%\bibitem{born1}
%S. Jadach, B.F.L. Ward and Z. Was, Phys. Rev. D{\bf 63} (2001) 113009.
%\bibitem{van1}R. Hamberg, W. L. van Neerven and T Matsuura, 
%Nucl. Phys. {\bf B359} (1991) 343.
%\bibitem{van2}W.L. van Neerven and E.B. Zijlstra, Nucl.Phys. {\bf B382} (1992)
%11, and references therein.
%\bibitem{mrst1} A.D. Martin {\it et al.}, Phys. Rev. D{\bf 51} (1995) 4756.

%\bibitem{ja}
%C Jarlskog in {\it CP Violation}, ed. C Jarlskog
%(World Scientific, Singapore, 1988).

%\bibitem{ma}
%L. Maiani, {\it Phys. Lett.} B {\bf 62}, 183 (1976).

%\bibitem{bu}
%J.D. Bjorken and I. Dunietz, {\it Phys. Rev.} D {\bf 36}, 2109 (1987).

%\bibitem{Marnelius} R. Marnelius, {\it Acta Phys. Pol.} 
%{\bf B13}, 669 (1982).

%\bibitem{Bjorken} J. D. Bjorken, in {\it Lecture Notes on Current-Induced 
%Reactions}, ed. J. Komer {\it et al.} (Springer, Heidelberg, 1975).

%\bibitem{Bohr} A. Bohr and B. R. Mottelson, {\it Nuclear Structure} 
%(Benjamin, New York, 1969), Vol.~1, pp.~100--102.

%\bibitem{Webb} R. C. Webb, PhD thesis, Princeton University, 1972.

%\bibitem{Toimela} T. Toimela, Helsinki Research Institute for 
%Theoretical Physics, Report No. HU-TFT-82-37, 1982 (unpublished).
\end{thebibliography}