[Rivet-svn] r3856 - schools/2012-CERN/handouts/day2

[blackhole at projects.hepforge.org]

Author: dgrell
Date: Wed Jul 25 08:04:26 2012
New Revision: 3856

Log:
Copy of handouts

Added:
   schools/2012-CERN/handouts/day2/Hw-day2.tex
      - copied unchanged from r3855, schools/2011-Kyoto/handouts/day2/Hw-day2.tex
   schools/2012-CERN/handouts/day2/Py-day2.tex
      - copied unchanged from r3855, schools/2011-Kyoto/handouts/day2/Py-day2.tex
   schools/2012-CERN/handouts/day2/Sh-day2.tex
      - copied unchanged from r3855, schools/2011-Kyoto/handouts/day2/Sh-day2.tex

Copied: schools/2012-CERN/handouts/day2/Hw-day2.tex (from r3855, schools/2011-Kyoto/handouts/day2/Hw-day2.tex)
==============================================================================
--- /dev/null	00:00:00 1970	(empty, because file is newly added)
+++ schools/2012-CERN/handouts/day2/Hw-day2.tex	Wed Jul 25 08:04:26 2012	(r3856, copy of r3855, schools/2011-Kyoto/handouts/day2/Hw-day2.tex)
@@ -0,0 +1,205 @@
+\documentclass[a4paper,10pt]{scrartcl}
+
+\usepackage{fullpage}
+\usepackage{amsmath}
+\usepackage{helvet}
+\usepackage{url}
+
+\setlength{\parindent}{0in}
+\newcommand{\done}{{\rm d}}
+\newcommand{\nnb}{\nonumber}
+
+%opening
+\title{IPMU-YITP School 2011 Tutorials: \\ Day 2 Z+Jets with Herwig++}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+
+For today's session, you will be working in small groups to create data
+for $Z$+jets events. At the end of the tutorial you will combine your
+results and discuss them.
+
+To share your results, we have created some webspace where you can
+upload your files so that the whole group can access them:
+\url{http://users.hepforge.org/~hoeth/mcschool_X/}
+(where {\tt X} is your group number).
+
+\section{ME Level}
+
+\subsection{Physics}
+
+%needs more explanations
+The signal process in event generation is calculated perturbatively
+using matrix elements. In this section of the tutorial, we will look at
+the effects on observables of adding additional hard radiation in the
+matrix element to production of $Z$-bosons.
+
+\subsection{Running Herwig++}
+
+The setups for this section can be found in the folder \url{~/school/day2/herwig/ME}.
+
+\begin{verbatim}
+  cd school/day2/herwig/ME
+  ls
+\end{verbatim}
+
+You will find three different input files named {\tt TVT-0jet.dat,
+TVT-1jet.dat} and {\tt TVT-Powheg.dat}; they all load {\tt
+Base.in}. Take a look at the files. In
+particular, look at the matrix element setup, where either the regular
+$Z$ ME or the $Z$+jet ME is selected. The Powheg ME gives an NLO
+description of the same process. Also inspect the Handler settings, which
+disable the shower. We are running four different Rivet analyses, to
+compare the setups.
+Once you have checked the setup, run Herwig using the commands
+below. You can also share out longer runs among your group.
+\begin{verbatim}
+  Herwig++ read TVT-....in
+  Herwig++ run TVT-....run -N 50000
+\end{verbatim}
+Here you should replace {\tt TVT-...} with the name of the input file to
+run. After the runs, compare the process cross-section from the {\tt .out}
+files.
+
+\subsection{Plotting your Results}
+
+Rivet's histogram output is written to {\tt .aida} files.
+You can collect  results from  other  members  of your  group  by copying  all
+relevant  `.aida' files  to a  common directory.
+
+To plot your results, enter the following two commands:
+
+\begin{verbatim}
+  rivet-mkhtml *.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command \\ {\tt mupdf plots/CDF\_2008\_S7540469/d01-x01-y01.pdf} etc.
+
+Why is the agreement with data in the low $p_\perp$ region so poor?
+Can you see where the effect of multi-jet events shows up?
+
+For the 0-jet sample, one would naively expect the $p_\perp$ of the $Z$
+boson to be $0$ (why?). But as you should see, it does get a very soft $p_\perp$
+kick. Do you have any idea where that comes from?
+
+\section{ME/PS Merging}
+
+\subsection{Physics}
+
+Monte Carlo event generators generally rely on separating events into
+different stages. As mentioned above, the hard interaction is calculated
+perturbatively using the matrix element approach. However, the
+computational work required for this increases approximately factorially
+with the perturbation order, so it is not realistically possible to
+calculate high-multiplicity events using purely this method.
+
+The parton shower describes the soft and collinear emissions from final
+state partons by resumming the leading logarithmic terms. However, as
+the non-leading terms are neglected, the parton shower does not describe
+hard or wide-angled parton emission well.
+
+Therefore, the multi-jet phase space is separated into two regions, with
+the hard, wide-angled emissions described by the matrix element, and the
+soft, collinear emissions described by the parton shower. Herwig does
+not yet include a mechanism for merging arbitrary multi-jet matrix
+elements with the parton shower. This will be included in a coming
+release. To better simulate the leading jet, a hard ME correction is
+implemented for several processes, including the Drell-Yan process we
+are looking at.
+
+This tutorial section will give you the opportunity to compare the
+radiation patterns produced from tree-level matrix element calculations
+with the corresponding parton shower results, and to study the effect of
+the hard ME correction.
+
+
+\subsection{Running Herwig++}
+
+Within your group, decide which runs you will prepare.
+
+The setups can be found in the directory named \url{~/school/day2/herwig/shower}.
+All now have the full generation chain of
+ME--Shower--Hadronization--Decays activated. The two {\tt 0jet} files
+have the hard ME correction on and off respectively.
+
+  Take a look at the input files, and once you have checked the
+  setup, run Herwig using the commands below.
+\begin{verbatim}
+  Herwig++ read TVT-....in
+  Herwig++ run TVT-....run -N 10000
+\end{verbatim}
+
+\subsection{Plotting your Results}
+
+Collect results from other members of your group by copying all relevant `.aida' files to a common directory.
+
+To plot your results, enter the following two commands:
+
+\begin{verbatim}
+  rivet-mkhtml *.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command \\ {\tt mupdf plots/CDF\_2008\_S7540469/d01-x01-y01.pdf} etc.
+
+
+\section{QED Radiation}
+
+\subsection{Physics}
+
+As well as the QCD effects that produce jets, there are also QED effects
+from radiated photons. In this part of the tutorial, we are going to
+look at the effect of this QED radiation. In the YFS formalism used by
+Herwig, the external lepton lines are dressed with resummed collinear
+photon radiation. The hardest emission is corrected to the exact matrix
+element, but the cross section is not affected.
+
+\subsection{Running Herwig++}
+
+The two setups can be found in the folder {\tt \~{}/school/day2/herwig/QED}.
+Within your group, decide which setup each member will run.
+Check the input files to see where QED radiation is included, then
+run Herwig using the commands:
+
+\begin{verbatim}
+  Herwig++ read TVT-0jets-QED-....in
+  Herwig++ run TVT-0jets-QED-....run -N 10000
+\end{verbatim}
+
+where {\tt ...} is either `on' or `off'. In addition to the  {\tt
+  MC\_ZJETS} analysis we used earlier, we also include {\tt
+  MC\_ZJETS\_NOCLUS}. In {\tt MC\_ZJETS}, the QED radiation in a cone around the
+lepton has been accounted for in the $Z$ reconstruction, while in
+{\tt MC\_ZJETS\_NOCLUS} this radiation has been ignored. What difference do you
+expect?
+
+\subsection{Plotting your Results}
+
+To plot the {\tt MC\_ZJETS\_NOCLUS} results into the same histogram as
+the {\tt MC\_ZJETS} results, we need to save them in an .aida file on
+their own:
+
+\begin{verbatim}
+  ./extract_noclus.sh TVT-0jets-QED-off.aida TVT-0jets-QED-on.aida
+\end{verbatim}
+
+Now you can create the plots as usual:
+
+\begin{verbatim}
+  rivet-mkhtml *CLUS.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command\goodbreak {\tt mupdf plots/MC\_ZJETS/Z\_mass.pdf}
+
+What differences can you see between the runs and analyses in the different
+observables?
+
+\end{document}

Copied: schools/2012-CERN/handouts/day2/Py-day2.tex (from r3855, schools/2011-Kyoto/handouts/day2/Py-day2.tex)
==============================================================================
--- /dev/null	00:00:00 1970	(empty, because file is newly added)
+++ schools/2012-CERN/handouts/day2/Py-day2.tex	Wed Jul 25 08:04:26 2012	(r3856, copy of r3855, schools/2011-Kyoto/handouts/day2/Py-day2.tex)
@@ -0,0 +1,210 @@
+\documentclass[a4paper,10pt]{scrartcl}
+
+\usepackage{fullpage}
+\usepackage{amsmath}
+\usepackage{helvet}
+\usepackage{url}
+
+\setlength{\parindent}{0in}
+\newcommand{\done}{{\rm d}}
+\newcommand{\nnb}{\nonumber}
+
+%opening
+\title{IPMU-YITP School 2011 Tutorials: \\ Day 2 Z+Jets with Pythia}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+
+For today's session, you will be working in small groups to create data
+for $Z$+jets events. At the end of the tutorial you will combine your
+results and discuss them.
+
+To share your results, we have created some webspace where you can
+upload your files so that the whole group can access them:
+\url{http://users.hepforge.org/~hoeth/mcschool_X/}
+(where {\tt X} is your group number).
+
+\section{ME Level}
+
+\subsection{Physics}
+
+%needs more explanations
+The signal process in event generation is calculated perturbatively
+using matrix elements. In this section of the tutorial, we will look at
+the effects on observables of adding additional hard radiation in the
+matrix element to production of $Z$-bosons.
+
+\subsection{Running Pythia}
+
+The setups for this section can be found in the folder \url{~/school/day2/pythia/ME}.
+
+\begin{verbatim}
+  cd school/day2/pythia/ME
+  ls
+\end{verbatim}
+
+You will find two different input files named {\tt 0jet.cmnd} and {\tt
+1jet.cmnd}, take a look at them. In particular, look at the matrix
+element setup, where either the regular $Z$ ME or the $Z$+jet ME is
+selected. Also inspect the settings which disable the shower. In the
+file {\tt pythia\_rivet}, we are linking Pythia to four different Rivet
+analyses, to compare the setups.  Once you have checked the setup, run
+Pythia using the command below. The preset number of events should be
+adequate, but you can always share out longer runs
+among your group.
+\begin{verbatim}
+  ./pythia_rivet Xjet.cmnd
+\end{verbatim}
+Here you should replace {\tt X} with $0$ or $1$, to run the two
+options.
+
+\subsection{Plotting your Results}
+
+Rivet's histogram output is written to {\tt .aida} files.
+You can collect  results from  other  members  of your  group  by copying  all
+relevant  `.aida' files  to a  common directory.
+
+To plot your results, enter the following two commands:
+
+\begin{verbatim}
+  rivet-mkhtml *.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command \\ {\tt mupdf plots/CDF\_2008\_S7540469/d01-x01-y01.pdf} etc.
+
+Why is the agreement with data in the low $p_\perp$ region so poor?
+Can you see where the effect of multi-jet events shows up?
+
+For the 0-jet sample, one would naively expect the $p_\perp$ of the $Z$
+boson to be $0$ (why?). But as you should see, it does get a very soft $p_\perp$
+kick. Do you have any idea where that comes from?
+
+Try editing some of the flags inside the config files for other
+configurations. The next section will demonstrate the different shower
+components in more detail.
+
+\section{ME/PS Merging}
+
+\subsection{Physics}
+
+Monte Carlo event generators generally rely on separating events into
+different stages. As mentioned above, the hard interaction is calculated
+perturbatively using the matrix element approach. However, the
+computational work required for this increases approximately factorially
+with the perturbation order, so it is not realistically possible to
+calculate high-multiplicity events using purely this method.
+
+The parton shower describes the soft and collinear emissions from final
+state partons by resumming the leading logarithmic terms. However, as
+the non-leading terms are neglected, the parton shower does not describe
+hard or wide-angled parton emission well.
+
+Therefore, the multi-jet phase space is separated into two regions, with
+the hard, wide-angled emissions described by the matrix element, and the
+soft, collinear emissions described by the parton shower. Pythia does
+not yet include a mechanism for merging arbitrary multi-jet matrix
+elements with the parton shower. This will be included in a coming
+release. To better simulate the leading jet, a hard ME correction is
+implemented for several processes, including the Drell-Yan process we
+are looking at.
+
+This tutorial section will give you the opportunity to compare the
+radiation patterns produced from tree-level matrix element calculations
+with the corresponding parton shower results, and to study the effect of
+the hard ME correction.
+
+
+\subsection{Running Pythia}
+
+The setups can be found in the directory named \url{~/school/day2/pythia/shower}.
+Both now have the full generation chain of
+ME--Shower--Hadronization--Decays activated. Within your group, you
+should create variations of the {\tt 0jet.cmnd} file where you explore
+the different shower settings mentioned in section 6.
+Once you have checked the
+  setup, run Pythia using the same command as before:
+\begin{verbatim}
+  ./pythia_rivet Xjet.cmnd
+\end{verbatim}
+
+\subsection{Plotting your Results}
+
+Collect results from other members of your group by copying all relevant `.aida' files to a common directory.
+
+To plot your results, enter the following two commands:
+
+\begin{verbatim}
+  rivet-mkhtml *.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command \\ {\tt mupdf plots/CDF\_2008\_S7540469/d01-x01-y01.pdf} etc.
+
+
+\section{QED Radiation}
+
+\subsection{Physics}
+
+As well as the QCD effects that produce jets, there are also QED effects
+from radiated photons. In this part of the tutorial, we are going to
+look at the effect of this QED radiation. QED emissions are handled by
+shower evolution, in the same spirit as QCD ones, so that it is
+straightforward to combine the two for emissions off quarks. In the
+current study we let the $Z$ decay to leptons, and so the shower
+primarily involves the emission of photons. (The shower also allows a
+photon to branch to a lepton or quark pair, which in its turn can
+radiate further, but this is rare.) Thus the external lepton lines are
+dressed with resummed collinear photon radiation. The hardest emission
+is corrected to the exact matrix element, but the cross section is not
+affected.
+
+\subsection{Running Pythia}
+
+The two setups can be found in the folder {\tt \~{}/school/day2/pythia/QED}.
+Within your group, decide which setup each member will run.
+Please make sure to use electrons only or muons only in the run
+files, not both together.
+Check the input files to see where QED radiation is included, then
+run Pythia using the commands:
+
+\begin{verbatim}
+   ./pythia_rivet XYZ.cmnd
+\end{verbatim}
+
+Where {\tt XYZ} is either `noQED' or `QED'. In addition to the  {\tt
+  MC\_ZJETS} analysis we used earlier, we also include {\tt
+  MC\_ZJETS\_NOCLUS}. In {\tt MC\_ZJETS}, the QED radiation in a cone around the
+lepton has been accounted for in the $Z$ reconstruction, while in
+{\tt MC\_ZJETS\_NOCLUS} this radiation has been ignored. What difference do you
+expect?
+
+\subsection{Plotting your Results}
+
+To plot the {\tt MC\_ZJETS\_NOCLUS} results into the same histogram as
+the {\tt MC\_ZJETS} results, we need to save them in an .aida file on
+their own:
+
+\begin{verbatim}
+  ./extract_noclus.sh noQED.aida QED.aida
+\end{verbatim}
+
+Now you can create the plots as usual:
+
+\begin{verbatim}
+  rivet-mkhtml *CLUS.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command {\tt mupdf plots/MC\_ZJETS/Z\_mass.pdf}
+
+What differences can you see between the runs and analyses in the different
+observables?
+
+\end{document}

Copied: schools/2012-CERN/handouts/day2/Sh-day2.tex (from r3855, schools/2011-Kyoto/handouts/day2/Sh-day2.tex)
==============================================================================
--- /dev/null	00:00:00 1970	(empty, because file is newly added)
+++ schools/2012-CERN/handouts/day2/Sh-day2.tex	Wed Jul 25 08:04:26 2012	(r3856, copy of r3855, schools/2011-Kyoto/handouts/day2/Sh-day2.tex)
@@ -0,0 +1,215 @@
+\documentclass[a4paper,10pt]{scrartcl}
+
+\usepackage{fullpage}
+\usepackage{amsmath}
+\usepackage{helvet}
+\usepackage{url}
+
+\setlength{\parindent}{0in}
+\newcommand{\done}{{\rm d}}
+\newcommand{\nnb}{\nonumber}
+
+%opening
+\title{IPMU-YITP School 2011 Tutorials: \\ Day 2 Z+Jets with Sherpa}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+
+For today's session, you will be working in small groups to create data
+for $Z$+jets events. At the end of the tutorial you will combine your
+results and discuss them.
+
+To share your results, we have created some webspace where you can
+upload your files so that the whole group can access them:
+\url{http://users.hepforge.org/~hoeth/mcschool_X/}
+(where {\tt X} is your group number).
+
+\section{ME Level}
+
+\subsection{Physics}
+
+%needs more explanations
+The signal process in event generation is calculated perturbatively
+using matrix elements. In this section of the tutorial, we will look at
+the effects on observables of adding additional hard radiation in the
+matrix element to production of $Z$-bosons.
+
+\subsection{Running Sherpa}
+
+The setups for this section can be found in the folder \url{~/school/day2/sherpa/ME}.
+
+\begin{verbatim}
+  cd school/day2/sherpa/ME
+  ls
+\end{verbatim}
+
+You will find four different run cards named {\tt Run.Xjet.dat}, where
+'X' is between 0 and 3. Take a look at the run cards. In
+particular, look at the (processes) section to see the number of additional jets
+produced by the matrix element. Also inspect the shower settings, which
+basically disable the shower.
+The cross sections have already been integrated for you. The results are in the
+folders {\tt Results.Xjet}. These are already set in the run cards.
+
+Once you are satisfied, run Sherpa and Rivet using the commands below.
+\begin{verbatim}
+  Sherpa RUNDATA=Run.Xjet.dat EVENTS=100000
+\end{verbatim}
+
+where the `X' in {\tt Xjet} is the relevant number between 0 and 3.
+
+\subsection{Plotting your Results}
+
+Rivet's histogram output is written to {\tt .aida} files.
+You can collect  results from  other  members  of your  group  by copying  all
+relevant  `.aida' files  to a  common directory.
+
+To plot your results, enter the following two commands:
+
+\begin{verbatim}
+  rivet-mkhtml Analysis.3jet.aida Analysis.2jet.aida Analysis.1jet.aida Analysis.0jet.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command \\ {\tt mupdf plots/CDF\_2008\_S7540469/d01-x01-y01.pdf} etc.
+
+Why is the agreement with data in the low $p_\perp$ region so poor?
+Can you see where the effect of multi-jet events shows up?
+
+For the ``max0jet'' sample, one would naively expect the $p_\perp$ of the $Z$
+boson to be $0$ (why?). But as you should see, it does get a very soft $p_\perp$
+kick. Do you have any idea where that comes from?
+
+\section{ME/PS Merging}
+
+\subsection{Physics}
+
+Monte Carlo event generators generally rely on separating events into
+different stages. As mentioned above, the hard interaction is calculated
+perturbatively using the matrix element approach. However, the
+computational work required for this increases approximately factorially
+with the perturbation order, so it is not realistically possible to
+calculate high-multiplicity events using purely this method.
+
+The parton shower describes the soft and collinear emissions from final
+state partons by resumming the leading logarithmic terms. However, as
+the non-leading terms are neglected, the parton shower does not describe
+hard or wide-angled parton emission well.
+
+Therefore, the multi-jet phase space is separated into two regions, with
+the hard, wide-angled emissions described by the matrix element, and the
+soft, collinear emissions described by the parton shower. Sherpa employs
+a procedure called CKKW merging to combine the matrix elements with the
+parton shower, while avoiding double-counting of phase space, and
+minimizing the dependence on the choice of phase space cut.
+
+This tutorial section will give you the opportunity to compare the
+radiation patterns produced from tree-level matrix element calculations
+with the corresponding parton shower results, by comparing results of
+event generation with a maximum of 0, 1, 2, or 3 jets in the matrix
+element.
+
+{\bf Please note:}
+{\it The comparison of results for different numbers of jets in the
+matrix element is just an exercise for this tutorial. When using ME/PS
+merging it is always advisable to have as many jets as computationally
+possible in the matrix element.}
+
+
+\subsection{Running Sherpa}
+
+Within your group, decide which jet multiplicities each member will run.
+Remember that the higher the jet multiplicity, the longer the run will take.
+
+The setups can be found in the directory named \url{~/school/day2/sherpa/merging}.
+
+Take a look at the run cards. In particular, look at the (processes) section,
+and check that you are generating events with the correct number of jets in the
+final state.
+
+Once you are satisfied, run Sherpa and Rivet using the command below.
+
+\begin{verbatim}
+  Sherpa RUNDATA=Run.Xjet.dat EVENTS=100000
+\end{verbatim}
+
+\noindent where the `X' in {\tt Xjet} is the relevant number between 0 and 3.
+
+\subsection{Plotting your Results}
+
+Collect results from other members of your group by copying all relevant `.aida' files to a common directory.
+
+To plot your results, enter the following two commands:
+
+\begin{verbatim}
+  rivet-mkhtml Analysis.3jet.aida Analysis.2jet.aida Analysis.1jet.aida Analysis.0jet.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command \\ {\tt mupdf plots/CDF\_2008\_S7540469/d01-x01-y01.pdf} etc.
+
+
+\section{QED Radiation}
+
+\subsection{Physics}
+
+As well as the QCD effects that produce jets, there are also QED effects
+from radiated photons. In this part of the tutorial, we are going to
+look at the effect of this QED radiation. In the YFS formalism used by
+Sherpa, the external lepton lines are dressed with resummed soft photon
+radiation. The hardest emission is corrected to the exact matrix
+element, but the cross section is not affected.
+
+\subsection{Running Sherpa}
+
+The two setups can be found in the folder {\tt \~{}/school/day2/sherpa/QED}.
+Within your group, decide which setup each member will run.
+Since we are not asking for extra jets this time, your job has to
+integrate the cross section.
+
+Take a look at the run cards. In particular, check in the (me) section to
+see if QED radiation is included.
+
+Run Sherpa and Rivet using the command:
+
+\begin{verbatim}
+  Sherpa RUNDATA=Run.*.dat EVENTS=100000
+\end{verbatim}
+
+where the * is either `ME' or `Off'.
+Here we are running two pseudo-analyses (i.e. without reference data), on the
+one hand to compare the runs, and on the other hand to compare two different
+analyses: In {\tt MC\_ZJETS}, the QED radiation in a cone around the
+lepton has been accounted for in the $Z$ reconstruction, while in
+{\tt MC\_ZJETS\_NOCLUS} this radiation has been ignored. What difference do you
+expect?
+
+\subsection{Plotting your Results}
+
+To plot the {\tt MC\_ZJETS\_NOCLUS} results into the same histogram as
+the {\tt MC\_ZJETS} results, we need to save them in an .aida file on
+their own:
+
+\begin{verbatim}
+  ./extract_noclus.sh Analysis.Off.aida Analysis.ME.aida
+\end{verbatim}
+
+Now you can create the plots as usual:
+
+\begin{verbatim}
+  rivet-mkhtml *CLUS.aida
+  firefox plots/index.html &
+\end{verbatim}
+
+\noindent You can also look at the .pdf files directly using, for example, the
+command {\tt mupdf plots/MC\_ZJETS/Z\_mass.pdf}
+
+What differences can you see between the runs and analyses in the different
+observables?
+
+\end{document}