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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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\end{document}

