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\begin{document}
\Title{Status of the EW cross sections for the LHC}
\bigskip\bigskip
%+\addtocontents{toc}{{\it D. Reggiano}}
%+\label{ReggianoStart}
\begin{raggedright}
{\it G. Balossini, G. Montagna \\
Dipartimento di Fisica Nucleare e Teorica, Universit\`a di Pavia \\
and INFN, Sezione di Pavia, Italy \\
C.M. Carloni Calame \\
INFN, Frascati (Italy) \\
and School of Physics and Astronomy,
Southampton University, Southampton (UK) \\
O. Nicrosini, F. Piccinini\index{Piccinini, F.} \\
INFN, Sezione di Pavia, Italy \\
A. Vicini \\
Dipartimento di Fisica, Universit\`a di Milano and INFN,
Sezione di Milano, Italy}
\bigskip\bigskip
\end{raggedright}
\section{Introduction}
Charged and neutral current Drell-Yan (D-Y) processes,
{\emph{ i.e.}} $p p\hspace{-8pt}{}^{{}^{(-)}} \to W\to l \nu_l + X$,
and $p p\hspace{-8pt}{}^{{}^{(-)}} \to Z/\gamma \to l^+ l^- + X$
play a very important role at hadron colliders, since they
have huge cross sections,
(\emph{e.g.} $\sigma(p p \to W\to l \nu_l + X)$ $\sim$ $20$~nb at LHC and
about a factor of ten less for
$\sigma(p p \to Z/\gamma \to l^+ l^- + X)$) and
are easily detected, given the presence of at least a high $p_\perp$ lepton,
which to trigger on.
Present analysis by ATLAS and CMS of early LHC data
attained single-$W$ cross section measurements with an accuracy of the order
of few per cent.
For these reasons and also because the physics around $W$ and $Z$
mass scale is presently known with high precision after the LEP and Tevatron
experience, D-Y processes provide standard candles for detector
calibration. Moreover, single-$W$ as signal by itself
will allow to perform a precise measurement of the $W$ mass with a
foreseen final uncertainty of the order of 15~MeV at LHC
(20~MeV at Tevatron), a very important
ingredient for precision tests of the Standard Model, when associated with
a top mass uncertainty of the order of 1-2~GeV.
Also, from the forward-backward asymmetry of the charged lepton pair in
$p p \to Z/\gamma \to e^+ e^-$ the mixing angle $\sin^2 \vartheta_W$ could
be extracted with a precision of $1 \times 10^{-4}$.
Last, single-$W$ and single-$Z$ processes provide important
observables for new physics searches: in fact
the high tail of the $l^+ l^-$ invariant mass and of the $W$ transverse
mass is sensitive to the presence of extra gauge bosons predicted
in many extension of the Standard Model, which
could lie in the TeV energy scale detectable at LHC.
An important observation is that about 50\% of the total present
systematic error at LHC on the D-Y cross section measurement
is due to theoretical uncertainties, in addition to the uncertainty
due to PDF's knowledge.
The sources of uncertainty in the theoretical predictions
are essentially of perturbative and non-perturbative origin.
The latter ones comprise the uncertainties related to the parton
distribution functions and power corrections
to resummed differential cross sections,
which will not be discussed here.
In the following we review the current state-of-the-art
on the calculation of higher order QCD and electroweak (EW)
radiative corrections and their implementation in simulation tools, and we present some recent results about the
combination of QCD and EW corrections to $W$ production
at the LHC.
\section{Status of theoretical calculations and tools}
In the present section, a sketchy summary of the main computational tools for EW gauge boson
production at hadron colliders is presented.
Concerning QCD calculations and tools, the present situation reveals quite a rich structure,
that includes
next-to-leading-order (NLO) and next-to-next-to-leading-order (NNLO)
corrections to $W/Z$ total production rate~\cite{AEM,HvNM},
NLO calculations for $W, Z + 1, 2 \, \, {\rm jets}$
signatures (available in the codes DYRAD and MCFM)~\cite{GGK,MCFM},
resummation of leading and next-to-leading logarithms due to soft gluon
radiation (implemented in the Monte Carlo ResBos)~\cite{BY,resbos},
NLO corrections merged with QCD Parton Shower (PS)
evolution (in the event generators
MC@NLO and POWHEG)~\cite{MC@NLO, POWHEG},
NNLO corrections to $W/Z$ production in fully differential
form (available in the Monte Carlo programs FEWZ)~\cite{mp} and
DYNNLO~\cite{dynnlo}. Very recently the NNLL resummation of
$W/Z$ transverse momentum appeared in the literature~\cite{qtnnll}.
As far as complete ${\cal O}(\alpha)$ EW corrections
to D-Y processes
are concerned, they have been computed independently by various
authors in~\cite{dk, bw,ZYK,SANC,CMNV} for $W$ production and
in~\cite{zgrad2, Zykunov2007,HORACEZ, SANCZ,DH} for $Z$ production.
EW tools implementing exact NLO corrections to $W/Z$
production are DK~\cite{dk}, WGRAD2~\cite{bw}, ZGRAD2~\cite{zgrad2},
SANC~\cite{SANC}
and HORACE~\cite{CMNV}, HORACE~\cite{HORACEZ}.
From the calculations above, it turns out that NLO EW
corrections are dominated,
in the resonant region, by final-state QED radiation containing
large collinear logarithms
of the form $\log(\hat{s}/m_l^2)$, where $\hat{s}$ is the squared
partonic centre-of-mass (c.m.) energy
and $m_l$ is the lepton mass. Since these corrections amount to
several per cents around the
jacobian peak of the $W$ transverse mass and lepton transverse momentum
distributions and cause a
significant shift (of the order of 100-200~MeV) in the extraction of the
$W$ mass $M_W$ at the Tevatron,
the contribution of higher-order corrections due to multiple photon
radiation from the
final-state leptons must be taken into account in the theoretical
predictions, in view of the expected precision in the $M_W$
measurement at the LHC.
The contribution due to multiple photon radiation has been computed,
by means of a QED PS approach~\cite{CMNT}
and implemented in the event generator HORACE. An independent approach
is followed in WINHAC~\cite{winhac}, where the multiple photon radiation
is described with the YFS exponentiation formalism. The exact
${\cal O}(\alpha)$ contribution is obtained
by means of an interface to the SANC system~\cite{bbjkp}.
Higher-order QED contributions to $W$ production have been calculated independently in~\cite{DK2007} within the collinear structure functions approach
It is worth noting that, for what concerns the precision
measurement of $M_W$, the shift induced by higher-order QED corrections is about
10\% of that caused by one-photon emission and of opposite sign, as shown in
~\cite{CMNT}. Therefore, such an effect is non-negligible in view of the aimed accuracy
in the $M_W$ measurement at the LHC.
A further important phenomenological feature of EW corrections is that, in the
region important for new physics searches (i.e. where the $W$ transverse mass is much
larger than the $W$ mass or the invariant mass of the final state leptons is much larger
than the $Z$ mass), the NLO EW effects become large (of the order of
20-30\%) and negative, due to the appearance of EW Sudakov logarithms
$\propto - (\alpha/\pi) \log^2 ({\hat s}/M_V^2)$, $V = W,Z$~\cite{dk,bw,CMNV,zgrad2,Zykunov2007,HORACEZ}.
In spite of this detailed knowledge of higher-order EW and QCD
corrections, the combination of their effects is presently under investigation.
Some attempts have been explored in the
literature~\cite{cy, ward2007, jadach2007}.
Many analysis at the resonance peak rely on the LL factorized approach
where a QCD Parton Shower is interfaced to PHOTOS~\cite{photos}
(or SOPHTY~\cite{sophty} as in the
case of HERWIG++) for the simulation of final state QED radiation.
Preliminary tests of the level of precision for this kind of approach
has been discussed at the $W/Z$ peak, for LHC energies
of 7, 10 and 14 TeV~\cite{ahy}.
Recent activity is devoted to the inclusion of NLO EW corrections
in the framework of a QCD generator, with possible inclusion of
higher order QED corrections~\cite{rssss,bcmmnpt}.
Here our approach is discussed in some detail~\cite{bcmmnpt}.
A first strategy for the combination of EW and QCD
corrections consists in the
following formula
\begin{eqnarray}
\left[\frac{d\sigma}{d\cal O}\right]_{{\rm QCD} \& {\rm EW}} =
\left\{\frac{d\sigma}{d\cal O}\right\}_{{\rm MC@NLO}}
+\left\{\left[\frac{d\sigma}{d\cal O}\right]_{{\rm EW}} -
\left[\frac{d\sigma}{d\cal O}\right]_{{\rm Born}} \right\}_{{\rm HERWIG\, \, PS}}
\label{eq:qcd-ew}
\end{eqnarray}
where ${d\sigma/d\cal O}_{{\rm MC@NLO}}$ stands for the prediction of the
observable ${d\sigma/d\cal O}$
as obtained by means of MC@NLO,
${d\sigma/d\cal O}_{{\rm EW}}$ is the HORACE
prediction for the EW corrections to the ${d\sigma/d\cal O}$ observable,
and ${d\sigma/d\cal O}_{{\rm Born}}$ is the lowest-order result
for the observable of interest. The label {{\rm HERWIG PS} in the second term
in r.h.s.
of Eq. (\ref{eq:qcd-ew}) means that EW corrections are convoluted with QCD PS
evolution through the HERWIG event generator, in order to (approximately)
include
mixed ${\cal O}(\alpha \alpha_s)$ corrections and to obtain a more realistic
description of the observables under study. In Eq.~(\ref{eq:qcd-ew})
the infrared part of QCD corrections is factorized, whereas the infrared-safe
matrix element residue is included in an additive form. It is otherwise
possible to implement a fully factorized combination
(valid for infrared safe observables) as follows:
\begin{eqnarray}
\left[\frac{d\sigma}{d\cal O}\right]_{{\rm QCD} \otimes {\rm EW}} = & &
\left( 1 + \frac{
\left[{d\sigma} / {d\cal O}\right]_{\rm MC@NLO} - \left[{d\sigma}/{d\cal O}\right]_{\rm HERWIG\, \, PS}
}{
\left[{d\sigma}/{d\cal O}\right]_{\rm LO/NLO}
}
\right) \times \nonumber \\
& & \times
\left\{
\frac{d\sigma}{{d\cal O}_{\rm EW}}
\right\}_{{\rm HERWIG\, \, PS}} ,
\label{eq:qcd-ew-factor}
\end{eqnarray}
where the ingredients are the same as in Eq.~(\ref{eq:qcd-ew}) but also the QCD matrix element residue is now factorized.
It is worth noticing that the QCD correction factor in front of $\left\{
d\sigma / {d\cal O}_{\rm EW}
\right\}_{{\rm HERWIG\, \, PS}}
$
is defined in terms of two different normalization cross sections,
namely the LO or the NLO one, respectively. The two prescriptions
differ at order $\alpha_s^2$ by non-leading
contributions. Nevertheless, Eq.~(\ref{eq:qcd-ew-factor}) normalized
in terms of the LO cross section can give rise to pathologically
large order $\alpha_s^2$ corrections in the presence of huge NLO effects.
On the other hand, when NLO matrix element effects do not introduce
particularly relevant corrections,
the two prescriptions are substantially equivalent.
Eqs.~(\ref{eq:qcd-ew}) and ~(\ref{eq:qcd-ew-factor}) have the very
same ${\cal O}(\alpha)$ and ${\cal O}(\alpha_s)$ content, differing
by terms at the order $\alpha \alpha_s$. Their relative difference
can be taken as an estimate of the uncertainty of QCD \& EW combination.
In ~\cite{bcmmnpt} a complete numerical study has been performed on the
various approximations of the radiative corrections to D-Y and for
several physical observables, for two standard
event selections at Tevatron on peak (Tevatron)
and LHC on peak (LHC a) and far off peak (LHC b).
We summarize in Table~\ref{tab:deltafinal}
the relative effects of the different sources
of corrections to the integrated cross section.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\delta(\%)$ & NLO QCD & NLL QCD & NLO EW & Shower QCD & $O(\alpha \alpha_s)$ \\
\hline
Tevatron & 8 &16.8 & -2.6 & -1.3 & $\sim 0.5$ \\
\hline
LHC a & -2 & 12.4 & -2.6 & 1.4 & $\sim 0.5$ \\
\hline
LHC b & 21.8 & 20.9 & -21.9 & -0.6 & $\sim 5$ \\
\hline
\end{tabular}
\caption{Relative effect of the main sources of QCD, EW and mixed radiative corrections to the integrated cross sections
for the Tevatron, LHC a and LHC b.}
\label{tab:deltafinal}
\end{center}
\end{table}
NLO QCD is the complete $O(\alpha_s)$ correction,
NLL QCD is the matrix element contribution of the NLO QCD correction,
NLO EW is the full $O(\alpha)$ correction,
Shower QCD stands for the $O(\alpha_s^n), n \geq 2$
correction and $O(\alpha \alpha_s)$
represents the mixed EW-QCD corrections estimated by properly
combining the additive and factorized cross sections.
It is worth noticing in particular that the latter corrections
remain below the 1\% level for typical event selections at the Tevatron
and the LHC, while they can amount to some per cent in the region
important for new physics searches at the LHC.
\section{Conclusions}
After reviewing the currently available theoretical calculations and tools,
we discussed a strategy to estimate the missing higher order corrections to
D-Y processes.
In general, for the LHC we remarked that available calculations and tools do not currently allow to reach a
theoretical accuracy better than some per cent level, when excluding PDF uncertainties. Future measurements at the LHC would require the consistent inclusion
of NLO EW matched with multiphoton radiation within a unified NLO
QCD generator. In a longer run
probably require the
calculation of complete ${\cal O} (\alpha\alpha_s)$ corrections.
Recent work in this direction is the calculation of the two-loop
${\cal O}(\alpha \alpha_s)$ virtual corrections to
D-Y production \cite{ks}
and of the one loop ${\cal O}(\alpha)$ corrections to the
signature $W + 1$~jet \cite{ddkm}.
\bigskip
F.P. is grateful to C. Grojean and to G. Isidori for their kind invitation.
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\end{document}