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\begin{document}
\Title{Status of Monte-Carlo event generators}
\bigskip\bigskip
\begin{raggedright}
{\it Stefan Hoeche}\hfill SLAC-PUB-15266\\
{\it SLAC National Accelerator Laboratory\\
Menlo Park, CA 94025, USA}
\bigskip\bigskip
\end{raggedright}
A dominant part in the analysis of early LHC data is played by general-purpose Monte-Carlo
(MC) event generators~\cite{Webber:1986mc,*Buckley:2011ms}, which are employed by both
experimentalists and theorists to obtain particle level predictions for collider
experiments. Hundreds of final-state particles are typically produced in LHC collisions,
and the reactions involve both large and small momentum transfer. The high-dimensional
phase space and the non-abelian, nonlinear nature of Quantum Chromodynamics (QCD) make
an exact solution of the problem impossible. Instead, MC event generators resort
to factorization, which allows to split events into different stages, ordered descending
in invariant momentum transfer. In this picture, a hard interaction, described through
fixed-order perturbation theory, is followed by multiple Bremsstrahlung emissions
off initial- and final-state particles and, eventually, by the hadronization process.
Each step is simulated independently.
Three general-purpose Monte-Carlo event generators are currently available which
implement this paradigm: HERWIG~\cite{Corcella:2000bw,*Bahr:2008pv},
Pythia~\cite{Sjostrand:2006za,*Sjostrand:2007gs} and Sherpa~\cite{Gleisberg:2003xi,
*Gleisberg:2008ta}. A comprehensive description of the physics models implemented
in these programs can be found in~\cite{Webber:1986mc,*Buckley:2011ms}.
Traditionally, multi-purpose event generators compute hard processes at the
lowest order in the perturbative expansion. This approximation leads to serious deficiencies
in the description of final states with large jet multiplicity.
Tree-level matrix element generators have therefore been constructed,
which can cope with arbitrary final-states. The most widely
used programs nowadays are ALPGEN, AMEGIC, Comix, HELAC and MadGraph~\cite{Mangano:2002ea,
*Krauss:2001iv,*Gleisberg:2008fv,*Draggiotis:2002hm,*Stelzer:1994ta}.
Parton-level events produced by these tools are processed by general-purpose event generators
to implement parton showers and hadronization. Although independent programs in principle,
matrix-element generators like the above should thus be viewed as an integral part of the
simulation chain in general-purpose programs. Their extension to new physics scenarios is
handled by FeynRules~\cite{Christensen:2008py},
a Mathematica package, which automatically derives interaction vertices from
virtually arbitrary Lagrangians.
Predictions for observables in multi-jet final states involve high powers of the
strong coupling, and thus, they have large associated uncertainties. It is therefore
desirable to improve the description of high-multiplicity events through
next-to-leading order (NLO) calculations. Real and virtual NLO corrections can be
combined in an automated way using universal infrared subtraction algorithms~\cite{
Frixione:1995ms,*Catani:1996vz,*Catani:2002hc},
which are implemented in various tree-level matrix-element generators\cite{
Gleisberg:2007md,*Frederix:2008hu,*Frederix:2009yq,*Czakon:2009ss}.
The computation of many challenging background processes at the LHC was
accomplished with the help of these tools. Prominent examples include
$pp\to W/Z$+4~jets, $pp\to$4jets and $pp\to t\bar{t}b\bar{b}$~\cite{
Berger:2010zx,*Ita:2011wn,*Bern:2011ep,*Bevilacqua:2009zn}.
A variety of processes can now be computed in a fully automated fashion by linking
the matrix element generators described above with dedicated programs for one-loop
virtual matrix elements through a standardized interface~\cite{Binoth:2010xt}.
Computing virtual corrections often poses the greatest challenge, both because of
complexity and numerical stability. Tremendous progress was made in this field,
leading to new computational algorithms based on generalized unitarity.
Automated calculations of one-loop corrections have since become available in the
BlackHat, GoSam, HelacNLO, MadLoop, OpenLoops and Rocket~\cite{Berger:2008sj,
*Cullen:2011ac,*vanHameren:2009dr,*Hirschi:2011pa,*KeithEllis:2009bu,Cascioli:2011va}
programs, as well as several others~\cite{Lazopoulos:2008ex,*Giele:2009ui,*LopezVal:2012ms}.
Additionally, more traditional, Feynman-diagram based techniques have been
extended and applied for example to the process
$pp\to W^+W^-b\bar{b}$~\cite{Denner:2005nn,*Binoth:2008uq,*Denner:2010jp}.
They are also used in the program OpenLoops~\cite{Cascioli:2011va}.
Table~\ref{tab:four_jets_nlo} shows an example of next-to-leading order
results for 4~jet production. The calculation was performed with BlackHat and Sherpa,
which exemplifies the possible synergy between programs for one-loop calculations
and leading order event generators.
\begin{table}
\scalebox{0.81}{
\begin{tabular}{||c||c|c|c|c|c||}
\hline
no. jets & ATLAS &LO &ME+PS & NLO & NLO+NP \\
\hline
\hline
$\geq 2$ & $620 \pm 1.3 {}^{+110}_{-66} \pm 24$ & $958(1)^{+316}_{-221}$
& $559 (5)$ & $1193(3)^{+130}_{-135} $ &
$ 1130(19)^{+124}_{-129}$
\\
\hline
$\geq 3$ & $43 \pm 0.13{}^{+12}_{-6.2}\pm 1.7$ & $93.4(0.1)^{+50.4}_{-30.3}$ &
$39.7 (0.9)$ & $54.5(0.5)^{+2.2}_{-19.9} $
& $50.2(2.1)^{+2.0}_{-18.3}$
\\
\hline
$\geq 4 $& $\; 4.3 \pm 0.04{}^{+1.4}_{-0.79} \pm 0.24\;$
& $\;9.98(0.01)^{+7.40}_{-3.95}\;$
& $\;3.97 (0.08)\;$ & $\;5.54(0.12)^{+ 0.08}_{-2.44}\;$
& $\;5.11 (0.29)^{+0.08}_{ -2.32} \;$
\\
\hline
\end{tabular} }
\caption{Total cross sections in nb for jet production at the LHC,
using the anti-$k_T$ jet algorithm with $R=0.4$. ATLAS are compared against LO, ME+PS
and NLO theoretical predictions. Numerical integration uncertainties
are given in parentheses, the scale dependence is quoted as
super- and subscripts. The last column gives NLO results including
non-perturbative corrections computed with Sherpa. Uncertainties shown with the
ATLAS data are statistical, jet energy scale, and detector unfolding.
Table taken from~\protect\cite{Berger:2010zx,*Ita:2011wn,*Bern:2011ep,*Bevilacqua:2009zn}.\label{tab:four_jets_nlo}}
\end{table}
While the production of jets in high-energy collisions is typically described
very well by fixed-order calculations, the modeling of inner jet structure in this approach
is poor. The composition of jets in terms of several partons should therefore be simulated
by parton showers, which employ collinear factorization properties of scattering
amplitudes to sum leading and certain subleading logarithmic corrections to hard
scattering processes. The difference between existing parton-shower implementations in
HERWIG~\cite{Gieseke:2003rz,*Hamilton:2006ms}, Pythia~\cite{Sjostrand:2004ef,*Corke:2009tk,*Corke:2010zj}
and Sherpa~\cite{Schumann:2007mg,*Hoeche:2009xc} lies in the parametrization of the
radiative phase space, the splitting functions which are employed and, in particular,
the splitting kinematics.
Sherpa implements a dipole-like parton shower~\cite{Schumann:2007mg,*Hoeche:2009xc},
which is based on the Catani-Seymour dipole subtraction method in the large-$N_c$ approximation.
The advantage compared to traditional parton showers is an improved description of
soft-collinear regions, which arises as a consequence of the dependence of the
splitting function on the kinematics of the spectator parton. Similar ideas are
implemented in HERWIG~\cite{Platzer:2009jq}. Within Pythia, recent development focused
on improved matching to hard processes at next-to-leading order and on incorporating
multiple scattering and rescattering effects into parton shower simulations~\cite{
Sjostrand:2004ef,*Corke:2009tk,*Corke:2010zj}. First attempts have been made
in HERWIG to include all possible color correlations into the parton
shower~\cite{Platzer:2012np,*Platzer:2012hp}.
\begin{figure}\centering
\includegraphics[width=0.52\textwidth]{H-pt_comp_fixsc_runsc.eps}\hfill
\includegraphics[width=0.45\textwidth]{tevatron_mjj_exclusive.eps}
\caption{Left: Transverse momentum of the Higgs boson in $h$+jet events at the LHC (7~TeV).
Results from POWHEG simulations with different scale choice are compared against each other
and against predictions from HqT~\cite{Bozzi:2005wk}. The simulation was performed by
combining MadGraph with MCFM and the POWHEG Box.
Right: Di-jet mass in $W$+2jet events at the Tevatron (1.96~TeV).
Results from aMC@NLO are compared against predictions from ALPGEN and against
an NLO calculation. Figures taken from~\protect\cite{Campbell:2012am,*Frederix:2011ig}.\label{fig:powheg}}
\end{figure}
Higher-order tree-level calculations and parton showers, as introduced above,
are two essentially complementary approaches to simulating perturbative QCD
interactions in general-purpose Monte-Carlo. It is desirable to combine both,
in order to obtain the best possible description of jet production and evolution.
To this end, two different strategies have been exploited, which are known
as matching and merging.
Matching algorithms either aim at replacing parton-shower splitting operators
with the ratio of complete higher-order matrix elements divided by the Born,
or they provide means to correct for the difference between the two. The main problem
to be solved is that parton showers alter the kinematics of partonic final states.
If the underlying parton-level calculation is performed at NLO, this implies that
the Born contribution times the parton shower leads to spurious terms of order
$\alpha_s$, which must be subtracted to avoid double counting that would spoil
the NLO accuracy. Two universally applicable methods to accomplish this task were
suggested in the past, which are dubbed MC@NLO and POWHEG~\cite{
Frixione:2002ik,*Nason:2004rx,*Frixione:2007vw}.
Both methods were applied to a variety of processes, using the event generation
frameworks of HERWIG and Pythia.
The MC@NLO method has also been automated in the aMC@NLO framework, based
on MadLoop and HERWIG~\cite{Frederix:2011qg}
In contrast to MC@NLO, the POWHEG technique does not depend on the
parton-shower algorithm, hence, independent implementations exist~\cite{Alioli:2010xd}.
Within Sherpa, the POWHEG and MC@NLO methods have been automated~\cite{Hoeche:2010pf,*Hoeche:2011fd}.
Figure~\ref{fig:powheg} displays results for Higgs boson plus jet and
$W$ boson plus two jets production, which are some of the most challenging
processes recently implemented using matching methods.
\begin{figure}\centering
\begin{minipage}{0.54\textwidth}
\includegraphics[width=\textwidth]{modmu.eps}
\end{minipage}\hfill
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{jetmulti_main.eps}
\end{minipage}
\caption{Left: Jade 3$\to$2-jet rate in $e^+e^-\to$hadrons.
The renormalization scale dependence of NL$^3$-merging with
2 and 3-parton processes described at NLO is shown in the
ratio plot. Figure taken from~\protect\cite{Lavesson:2008ah}.
Right: Jet multiplicity in $W$+jets events at the LHC.
The renormalization scale dependence of MEPS@NLO merging with
up to $W$+2 parton processes described at NLO is shown as
an orange band. Figure taken from~\cite{Gehrmann:2012yg,*Hoeche:2012yf}.\label{fig:mepsnlo}}
\end{figure}
To improve the description of hard QCD radiation by general-purpose event generators,
so-called merging algorithms were proposed in the context of the LEP physics
program~\cite{Andre:1997vh,*Catani:2001cc,*Lonnblad:2001iq},
and subsequently extended for hadron collisions~\cite{Mangano:2001xp,*Krauss:2002up,
*Lavesson:2007uu,*Mangano:2006rw,*Alwall:2007fs,*Lonnblad:2011xx}.
The aim of these techniques is to replace the parton-shower approximation
with fixed-order matrix elements for only those partons or parton ensembles,
which can be identified with experimentally observed jets.
Merging algorithms define an unambiguous way to separate the phase-space
of real parton emission into a soft and a hard regime. Soft regions, where higher-order
corrections must be resummed, but can be approximated, are filled by the parton shower.
Hard regions, where soft and collinear approximations are unsuitable, are filled
by fixed-order calculations. Since fixed-order calculations are inclusive,
they must be made exclusive using the parton-shower no-branching probability, commonly
referred to as the Sudakov factor. In this manner, double-counting of logarithmically
enhanced terms is avoided, while sub-leading logarithms and finite corrections
are correctly included in the hard domain.
An extension of the original merging approaches, which generically maintains the exact
logarithmic accuracy of the parton shower while respecting the phase-space separation cut,
was achieved in~\cite{Hoeche:2009rj,*Hamilton:2009ne}.
The first merging at the leading to next-to-leading order was been presented
in~\cite{Lavesson:2008ah}, in a method based on explicit
subtraction of the LO and NLO contributions from the parton shower. The technique introduced
in~\cite{Gehrmann:2012yg,*Hoeche:2012yf} is based instead on an extended modified subtraction
similar to MC@NLO, which is implemented using truncated vetoed parton showers in the spirit
of~\cite{Hoeche:2009rj}. The two existing implementations of a merging method at the NLO
in~\cite{Lavesson:2008ah} and~\cite{Gehrmann:2012yg,*Hoeche:2012yf} both indicate a substantial
reduction of theoretical uncertainties, as exemplified in Fig.~\ref{fig:mepsnlo}.
Monte-Carlo event generators have a variety of free parameters, which can be tuned
such that predictions better match experimental data. Many of these paraemters
are connected to fragmentation models and underlying-event simulation, or more general,
to models for non-perturbative QCD effects. The resulting parameter space can be
quite large, which makes it impossible to find an optimal solution by hand.
Two new tools have been developed recently, which attack these problems using a
generator-independent validation and tuning strategy. Rivet~\cite{Buckley:2010ar},
implements analyses from the LEP,
Tevatron and LHC experiments in a common framework and allows simultaneous tests
of Monte-Carlo output against all available collider data. Professor~\cite{Buckley:2009bj}
employs Rivet to semi-automatically find the best point in the parameter space
of the event generator.
In summary, modern general-purpose event generators are highly sophisticated tools for LHC
phenomenology. They often implement perturbative QCD calculations at
next-to-leading order in the strong coupling and they provide parton showers
to include resummation effects. Extensions of event generators allow them to become
a platform for testing new physics models and improved descriptions of perturbative QCD
in the same framework. Validation and tuning has been in the focus of interest during the
first years of LHC running and has been simplified by dedicated tools.
We are indebted to the members of the MCnet network for discussions and input.
Support from the US Department of Energy (contract DE--AC02--76SF00515),
and from the US LHC Theory Initiative (NSF contract PHY-0705682)
is gratefully acknowledged.
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