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\begin{document}
\tikzstyle{mynode}=[rectangle split,rectangle split parts=2, draw,rectangle split part fill={lightgray, none}]
\title{HEJ 2 developer manual}
\author{}
\maketitle
\tableofcontents
\newpage
\section{Overview}
\label{sec:overview}
HEJ 2 is a C++ program and library implementing an algorithm to
apply \HIGHEJ resummation~\cite{Andersen:2008ue,Andersen:2008gc} to
pre-generated fixed-order events. This document is intended to give an
overview over the concepts and structure of this implementation.
\subsection{Project structure}
\label{sec:project}
HEJ 2 is developed under the \href{https://git-scm.com/}{git}
version control system. The main repository is on the IPPP
\href{https://gitlab.com/}{gitlab} server under
\url{https://gitlab.dur.scotgrid.ac.uk/hej/hej}. To get a local
copy, get an account on the gitlab server and use
\begin{lstlisting}[language=sh,caption={}]
git clone git@gitlab.dur.scotgrid.ac.uk:hej/hej.git
\end{lstlisting}
This should create a directory \texttt{hej} with the following
contents:
\begin{description}
\item[doc:] Contains additional documentation, see section~\ref{sec:doc}.
\item[include:] Contains the C++ header files.
\item[src:] Contains the C++ source files.
\item[t:] Contains the source code for the automated tests.
\item[CMakeLists.txt:] Configuration file for the \cmake build
system. See section~\ref{sec:cmake}.
\item[cmake:] Auxiliary files for \cmake. This includes modules for
finding installed software in \texttt{cmake/Modules} and templates for
code generation during the build process in \texttt{cmake/Templates}.
\item[config.yml:] Sample configuration file for running HEJ 2.
\item[events.lhe:] Fixed-order event sample that can be used as input
for HEJ 2.
\item[FixedOrderGen:] Contains the code for the fixed-order generator,
see section~\ref{sec:HEJFOG}.
\end{description}
In the following all paths are given relative to the
\texttt{hej} directory.
\subsection{Documentation}
\label{sec:doc}
The \texttt{doc} directory contains user documentation in
\texttt{doc/sphinx} and the configuration to generate source code
documentation in \texttt{doc/doxygen}.
The user documentation explains how to install and run HEJ 2. The
format is
\href{http://docutils.sourceforge.net/rst.html}{reStructuredText}, which
is mostly human-readable. Other formats, like \html, can be generated with the
\href{http://www.sphinx-doc.org/en/master/}{sphinx} generator with
\begin{lstlisting}[language=sh,caption={}]
make html
\end{lstlisting}
To document the source code we use
\href{https://www.stack.nl/~dimitri/doxygen/}{doxygen}. To generate
\html documentation, use the command
\begin{lstlisting}[language=sh,caption={}]
doxygen Doxyfile
\end{lstlisting}
in the \texttt{doc/doxygen} directory.
\subsection{Build system}
\label{sec:cmake}
For the most part, HEJ 2 is a library providing classes and
functions that can be used to add resummation to fixed-order events. In
addition, there is a relatively small executable program leveraging this
library to read in events from an input file and produce resummation
events. Both the library and the program are built and installed with
the help of \cmake.
Debug information can be turned on by using
\begin{lstlisting}[language=sh,caption={}]
cmake base/directory -DCMAKE_BUILD_TYPE=Debug
make install
\end{lstlisting}
This facilitates the use of debuggers like \href{https://www.gnu.org/software/gdb/}{gdb}.
The main \cmake configuration file is \texttt{CMakeLists.txt}. It defines the
compiler flags, software prerequisites, header and source files used to
build HEJ 2, and the automated tests.
\texttt{cmake/Modules} contains module files that help with the
detection of the software prerequisites and \texttt{cmake/Templates}
template files for the automatic generation of header and
source files. For example, this allows to only keep the version
information in one central location (\texttt{CMakeLists.txt}) and
automatically generate a header file from the template \texttt{Version.hh.in} to propagate this to the C++ code.
\subsection{General coding guidelines}
\label{sec:notes}
The goal is to make the HEJ 2 code well-structured and
readable. Here are a number of guidelines to this end.
\begin{description}
\item[Observe the boy scout rule.] Always leave the code cleaner
than how you found it. Ugly hacks can be useful for testing, but
shouldn't make their way into the main branch.
\item[Ask if something is unclear.] Often there is a good reason why
code is written the way it is. Sometimes that reason is only obvious to
the original author (use \lstinline!git blame! to find them), in which
case they should be poked to add a comment. Sometimes there is no good
reason, but nobody has had the time to come up with something better,
yet. In some places the code might just be bad.
\item[Don't break tests.] There are a number of tests in the \texttt{t}
directory, which can be run with \lstinline!make test!. Ideally, all
tests should run successfully in each git revision. If your latest
commit broke a test and you haven't pushed to the central repository
yet, you can fix it with \lstinline!git commit --amend!. If an earlier
local commit broke a test, you can use \lstinline!git rebase -i! if
you feel confident.
\item[Test your new code.] When you add some new functionality, also add an
automated test. This can be useful even if you don't know the
``correct'' result because it prevents the code from changing its behaviour
silently in the future. \href{http://www.valgrind.org/}{valgrind} is a
very useful tool to detect potential memory leaks.
\item[Stick to the coding style.] It is somewhat easier to read code
that has a uniform coding and indentation style. We don't have a
strict style, but it helps if your code looks similar to what is
already there.
\end{description}
\section{Program flow}
\label{sec:flow}
A run of the HEJ 2 program has three stages: initialisation,
event processing, and cleanup. The following sections outline these
stages and their relations to the various classes and functions in the
code. Unless denoted otherwise, all classes and functions are part of
the \lstinline!HEJ! namespace. The code for the HEJ 2 program is
in \texttt{src/main.cc}, all other code comprises the HEJ 2
library. Classes and free functions are usually implemented in header
and source files with a corresponding name, i.e. the code for
\lstinline!MyClass! can usually be found in
\texttt{include/HEJ/MyClass.hh} and \texttt{src/MyClass.cc}.
\subsection{Initialisation}
\label{sec:init}
The first step is to load and parse the \YAML configuration file. The
entry point for this is the \lstinline!load_config! function and the
related code can be found in \texttt{include/HEJ/YAMLreader.hh},
\texttt{include/HEJ/config.hh} and the corresponding \texttt{.cc} files
in the \texttt{src} directory. The implementation is based on the
\href{https://github.com/jbeder/yaml-cpp}{yaml-cpp} library.
The \lstinline!load_config! function returns a \lstinline!Config! object
containing all settings. To detect potential mistakes as early as
possible, we throw an exception whenever one of the following errors
occurs:
\begin{itemize}
\item There is an unknown option in the \YAML file.
\item A setting is invalid, for example a string is given where a number
would be expected.
\item An option value is not set.
\end{itemize}
The third rule is sometimes relaxed for ``advanced'' settings with an
obvious default, like for importing custom scales or analyses.
The information stored in the \lstinline!Config! object is then used to
initialise various objects required for the event processing stage
described in section~\ref{sec:processing}. First, the
\lstinline!get_analysis! function creates an object that inherits from
the \lstinline!Analysis! interface.\footnote{In the context of C++ the
proper technical expression is ``pure abstract class''.} Using an
interface allows us to decide the concrete type of the analysis at run
time instead of having to make a compile-time decision. Depending on the
settings, \lstinline!get_analysis! creates either a user-defined
analysis loaded from an external library (see the user documentation
\url{https://hej.web.cern.ch/HEJ/doc/current/user/}) or the default \lstinline!EmptyAnalysis!, which does
nothing.
Together with a number of further objects, whose roles are described in
section~\ref{sec:processing}, we also initialise the global random
number generator. We again use an interface to defer deciding the
concrete type until the program is actually run. Currently, we support the
\href{https://mixmax.hepforge.org/}{MIXMAX}
(\texttt{include/HEJ/Mixmax.hh}) and Ranlux64
(\texttt{include/HEJ/Ranlux64.hh}) random number generators, both are provided
by \href{http://proj-clhep.web.cern.ch/}{CLHEP}.
We also set up a \lstinline!LHEF::Reader! object (see
\href{http://home.thep.lu.se/~leif/LHEF/}{\texttt{include/LHEF/LHEF.h}}) for
reading events from a file in the Les
Houches event file format~\cite{Alwall:2006yp}. A small wrapper around
the
\href{https://www.boost.org/doc/libs/1_67_0/libs/iostreams/doc/index.html}{boost
iostreams} library allows us to also read event files compressed with
\href{https://www.gnu.org/software/gzip/}{gzip}. The wrapper code is in
\texttt{include/HEJ/stream.hh} and the \texttt{src/stream.cc}.
\subsection{Event processing}
\label{sec:processing}
In the second stage events are continously read from the event
file. After jet clustering, a number of corresponding resummation events
are generated for each input event and fed into the analysis and a
number of output files. The roles of various classes and functions are
illustrated in the following flow chart:
\begin{center}
\begin{tikzpicture}[node distance=2cm and 5mm]
\node (reader) [mynode]
{\lstinline!LHEF::Reader::readEvent!\nodepart{second}{read event}};
\node
(cluster) [mynode,below=of reader]
{\lstinline!Event! constructor\nodepart{second}{cluster jets}};
\node
(resum) [mynode,below=of cluster]
{\lstinline!EventReweighter::reweight!\nodepart{second}{perform resummation}};
\node
(cut) [mynode,below=of resum]
{\lstinline!Analysis::pass_cuts!\nodepart{second}{apply cuts}};
\node
(fill) [mynode,below left=of cut]
{\lstinline!Analysis::fill!\nodepart{second}{analyse event}};
\node
(write) [mynode,below right=of cut]
{\lstinline!CombinedEventWriter::write!\nodepart{second}{write out event}};
\node
(control) [below=of cut] {};
\draw[-{Latex[length=3mm, width=1.5mm]}]
(reader.south) -- node[left] {\lstinline!LHEF::HEPEUP!} (cluster.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(cluster.south) -- node[left] {\lstinline!Event!} (resum.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(resum.south) -- (cut.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(resum.south)+(7mm, 0cm)$) -- ($(cut.north)+(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(resum.south)-(7mm, 0cm)$) -- node[left] {\lstinline!Event!} ($(cut.north)-(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cut.south)-(3mm,0mm)$) .. controls ($(control)-(3mm,0mm)$) ..node[left] {\lstinline!Event!} (fill.east);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cut.south)-(3mm,0mm)$) .. controls ($(control)-(3mm,0mm)$) .. (write.west);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cut.south)+(3mm,0mm)$) .. controls ($(control)+(3mm,0mm)$) .. (fill.east);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cut.south)+(3mm,0mm)$) .. controls ($(control)+(3mm,0mm)$) ..node[right] {\lstinline!Event!} (write.west);
\end{tikzpicture}
\end{center}
The resummation is performed by the \lstinline!EventReweighter! class,
which is described in more detail in section~\ref{sec:resum}. The
\lstinline!CombinedEventWriter! writes events to zero or more output
files. To this end, it contains a number of objects implementing the
\lstinline!EventWriter! interface. These event writers typically write
the events to a file in a given format. We currently have the
\lstinline!LesHouchesWriter! for event files in the Les Houches Event
File format and the \lstinline!HepMCWriter! for the
\href{https://hepmc.web.cern.ch/hepmc/}{HepMC} format (Version 2 and 3).
\subsection{Resummation}
\label{sec:resum}
In the \lstinline!EventReweighter::reweight! member function, we first
classify the input fixed-order event (FKL, unordered, non-HEJ, \dots)
and decide according to the user settings whether to discard, keep, or
resum the event. If we perform resummation for the given event, we
generate a number of trial \lstinline!PhaseSpacePoint! objects. Phase
space generation is discussed in more detail in
section~\ref{sec:pspgen}. We then perform jet clustering according to
the settings for the resummation jets on each
\lstinline!PhaseSpacePoint!, update the factorisation and
renormalisation scale in the resulting \lstinline!Event! and reweight it
according to the ratio of pdf factors and \HEJ matrix elements between
resummation and original fixed-order event:
\begin{center}
\begin{tikzpicture}[node distance=2cm and 5mm]
\node (in) {};
\node (treat) [diamond,draw,below=of in,minimum size=3.5cm,
label={[anchor=west, inner sep=8pt]west:discard},
label={[anchor=east, inner sep=14pt]east:keep},
label={[anchor=south, inner sep=20pt]south:reweight}
] {};
\draw (treat.north west) -- (treat.south east);
\draw (treat.north east) -- (treat.south west);
\node
(psp) [mynode,below=of treat]
{\lstinline!PhaseSpacePoint! constructor};
\node
(cluster) [mynode,below=of psp]
{\lstinline!Event! constructor\nodepart{second}{cluster jets}};
\node
(gen_scales) [mynode,below=of cluster]
{\lstinline!ScaleGenerator::operator()!\nodepart{second}{update scales}};
\node
(rescale) [mynode,below=of gen_scales]
{\lstinline!PDF::pdfpt!,
\lstinline!MatrixElement!\nodepart{second}{reweight}};
\node (out) [below of=rescale] {};
\draw[-{Latex[length=3mm, width=1.5mm]}]
(in.south) -- node[left] {\lstinline!Event!} (treat.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(treat.south) -- node[left] {\lstinline!Event!} (psp.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(psp.south) -- (cluster.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(psp.south)+(7mm, 0cm)$) -- ($(cluster.north)+(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(psp.south)-(7mm, 0cm)$) -- node[left]
{\lstinline!PhaseSpacePoint!} ($(cluster.north)-(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(cluster.south) -- (gen_scales.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cluster.south)+(7mm, 0cm)$) -- ($(gen_scales.north)+(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cluster.south)-(7mm, 0cm)$) -- node[left]
{\lstinline!Event!} ($(gen_scales.north)-(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(gen_scales.south) -- (rescale.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(gen_scales.south)+(7mm, 0cm)$) -- ($(rescale.north)+(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(gen_scales.south)-(7mm, 0cm)$) -- node[left]
{\lstinline!Event!} ($(rescale.north)-(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(rescale.south) -- (out.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(rescale.south)+(7mm, 0cm)$) -- ($(out.north)+(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(rescale.south)-(7mm, 0cm)$) -- node[left]
{\lstinline!Event!} ($(out.north)-(7mm, 0cm)$);
\node (helper) at ($(treat.east) + (15mm,0cm)$) {};
\draw[-{Latex[length=3mm, width=1.5mm]}]
(treat.east) -- ($(treat.east) + (15mm,0cm)$)
-- node[left] {\lstinline!Event!} (helper |- gen_scales.east) -- (gen_scales.east)
;
\end{tikzpicture}
\end{center}
\subsection{Phase space point generation}
\label{sec:pspgen}
The resummed and matched \HEJ cross section for pure jet production of
FKL configurations is given by (cf. eq. (3) of~\cite{Andersen:2018tnm})
\begin{align}
\label{eq:resumdijetFKLmatched2}
% \begin{split}
\sigma&_{2j}^\mathrm{resum, match}=\sum_{f_1, f_2}\ \sum_m
\prod_{j=1}^m\left(
\int_{p_{j\perp}^B=0}^{p_{j\perp}^B=\infty}
\frac{\mathrm{d}^2\mathbf{p}_{j\perp}^B}{(2\pi)^3}\ \int
\frac{\mathrm{d} y_j^B}{2} \right) \
(2\pi)^4\ \delta^{(2)}\!\!\left(\sum_{k=1}^{m}
\mathbf{p}_{k\perp}^B\right)\nonumber\\
&\times\ x_a^B\ f_{a, f_1}(x_a^B, Q_a^B)\ x_b^B\ f_{b, f_2}(x_b^B, Q_b^B)\
\frac{\overline{\left|\mathcal{M}_\text{LO}^{f_1f_2\to f_1g\cdots
gf_2}\big(\big\{p^B_j\big\}\big)\right|}^2}{(\hat {s}^B)^2}\nonumber\\
& \times (2\pi)^{-4+3m}\ 2^m \nonumber\\
&\times\ \sum_{n=2}^\infty\
\int_{p_{1\perp}=p_{\perp,\mathrm{min}} }^{p_{1\perp}=\infty}
\frac{\mathrm{d}^2\mathbf{p}_{1\perp}}{(2\pi)^3}\
\int_{p_{n\perp}=p_{\perp,\mathrm{min}}}^{p_{n\perp}=\infty}
\frac{\mathrm{d}^2\mathbf{p}_{n\perp}}{(2\pi)^3}\
\prod_{i=2}^{n-1}\int_{p_{i\perp}=\lambda}^{p_{i\perp}=\infty}
\frac{\mathrm{d}^2\mathbf{p}_{i\perp}}{(2\pi)^3}\ (2\pi)^4\ \delta^{(2)}\!\!\left(\sum_{k=1}^n
\mathbf{p}_{k\perp}\right )\\
&\times \ \mathbf{T}_y \prod_{i=1}^n
\left(\int \frac{\mathrm{d} y_i}{2}\right)\
\mathcal{O}_{mj}^e\
\left(\prod_{l=1}^{m-1}\delta^{(2)}(\mathbf{p}_{\mathcal{J}_{l}\perp}^B -
\mathbf{j}_{l\perp})\right)\
\left(\prod_{l=1}^m\delta(y^B_{\mathcal{J}_l}-y_{\mathcal{J}_l})\right)
\ \mathcal{O}_{2j}(\{p_i\})\nonumber\\
&\times \frac{(\hat{s}^B)^2}{\hat{s}^2}\ \frac{x_a f_{a,f_1}(x_a, Q_a)\ x_b f_{b,f_2}(x_b, Q_b)}{x_a^B\ f_{a,f_1}(x_a^B, Q_a^B)\ x_b^B\ f_{b,f_2}(x_b^B, Q_b^B)}\ \frac{\overline{\left|\mathcal{M}_{\mathrm{HEJ}}^{f_1 f_2\to f_1 g\cdots
gf_2}(\{ p_i\})\right|}^2}{\overline{\left|\mathcal{M}_\text{LO, HEJ}^{f_1f_2\to f_1g\cdots
gf_2}\big(\big\{p^B_j\big\}\big)\right|}^{2}} \,.\nonumber
% \end{split}
\end{align}
The first two lines correspond to the generation of the fixed-order
input events with incoming partons $f_1, f_2$ and outgoing momenta
$p_j^B$, where $\mathbf{p}_{j\perp}^B$ and $y_j^B$ denote the respective
transverse momentum and rapidity. Note that, at leading order, these
coincide with the fixed-order jet momenta $p_{\mathcal{J}_j}^B$.
$f_{a,f_1}(x_a, Q_a),f_{b,f_2}(x_b, Q_b)$ are the pdf factors for the incoming partons with
momentum fractions $x_a$ and $x_b$. The square of the partonic
centre-of-mass energy is denoted by $\hat{s}^B$ and
$\mathcal{M}_\text{LO}^{f_1f_2\to f_1g\cdots gf_2}$ is the
leading-order matrix element.
The third line is a factor accounting for the different multiplicities
between fixed-order and resummation events. Lines four and five are
the integration over the resummation phase space described in this
section. $p_i$ are the momenta of the outgoing partons in resummation
phase space. $\mathbf{T}_y$ denotes rapidity
ordering and $\mathcal{O}_{mj}^e$ projects out the exclusive $m$-jet
component. The relation between resummation and fixed-order momenta is
fixed by the $\delta$ functions. The first sets each transverse fixed-order jet
momentum to some function $\mathbf{j_{l\perp}}$ of the resummation
momenta. The exact form is described in section~\ref{sec:ptj_res}. The second
$\delta$ forces the rapidities of resummation and fixed-order jets to be
the same. Finally, the last line is the reweighting of pdf and matrix
element factors already shown in section~\ref{sec:resum}.
There are two kinds of cut-off in the integration over the resummation
partons. $\lambda$ is a technical cut-off connected to the cancellation
of infrared divergencies between real and virtual corrections. Its
numerical value is set in
\texttt{include/HEJ/Constants.h}. $p_{\perp,\mathrm{min}}$ regulates
and \emph{uncancelled} divergence in the extremal parton momenta. Its
size is set by the user configuration \url{https://hej.web.cern.ch/HEJ/doc/current/user/HEJ.html#settings}.
It is straightforward to generalise eq.~(\ref{eq:resumdijetFKLmatched2})
to unordered configurations and processes with additional colourless
emissions, for example a Higgs or electroweak boson. In the latter case only
the fixed-order integration and the matrix elements change.
\subsubsection{Gluon Multiplicity}
\label{sec:psp_ng}
The first step in evaluating the resummation phase space in
eq.~(\ref{eq:resumdijetFKLmatched2}) is to randomly pick terms in the
sum over the number of emissions. This sampling of the gluon
multiplicity is done in the \lstinline!PhaseSpacePoint::sample_ng!
function in \texttt{src/PhaseSpacePoint.cc}.
The typical number of extra emissions depends strongly on the rapidity
span of the underlying fixed-order event. Let us, for example, consider
a fixed-order FKL-type multi-jet configuration with rapidities
$y_{j_f},\,y_{j_b}$ of the most forward and backward jets,
respectively. By eq.~(\ref{eq:resumdijetFKLmatched2}), the jet
multiplicity and the rapidity of each jet are conserved when adding
resummation. This implies that additional hard radiation is restricted
to rapidities $y$ within a region $y_{j_b} \lesssim y \lesssim
y_{j_f}$. Within \HEJ, we require the most forward and most backward
emissions to be hard \todo{specify how hard} in order to avoid divergences, so this constraint
in fact applies to \emph{all} additional radiation.
To simplify the remaining discussion, let us remove the FKL rapidity
ordering
\begin{equation}
\label{eq:remove_y_order}
\mathbf{T}_y \prod_{i=1}^n\int \frac{\mathrm{d}y_i}{2} =
\frac{1}{n!}\prod_{i=1}^n\int
\frac{\mathrm{d}y_i}{2}\,,
\end{equation}
where all rapidity integrals now cover a region which is approximately
bounded by $y_{j_b}$ and $y_{j_f}$. Each of the $m$ jets has to contain at least
one parton; selecting random emissions we can rewrite the phase space
integrals as
\begin{equation}
\label{eq:select_jets}
\frac{1}{n!}\prod_{i=1}^n\int [\mathrm{d}p_i] =
\left(\prod_{i=1}^{m}\int [\mathrm{d}p_i]\ {\cal J}_i(p_i)\right)
\frac{1}{n_g!}\prod_{i=m+1}^{m+n_g}\int [\mathrm{d}p_i]
\end{equation}
with jet selection functions
\begin{equation}
\label{eq:def_jet_selection}
{\cal J}_i(p) =
\begin{cases}
1 &p\text{ clustered into jet }i\\
0 & \text{otherwise}
\end{cases}
\end{equation}
and $n_g \equiv n - m$. Here and in the following we use the short-hand
notation $[\mathrm{d}p_i]$ to denote the phase-space measure for parton
$i$. As is evident from eq.~\eqref{eq:select_jets}, adding an extra emission
$n_g+1$ introduces a suppression factor $\tfrac{1}{n_g+1}$. However, the
additional phase space integral also results in an enhancement proportional
to $\Delta y_{j_f j_b} = y_{j_f} - y_{j_b}$. This is a result of the
rapidity-independence of the MRK limit of the integrand, consisting of the
matrix elements divided by the flux factor. Indeed, we observe that the
typical number of gluon emissions is to a good approximation proportional to
the rapidity separation and the phase space integral is dominated by events
with $n_g \approx \Delta y_{j_f j_b}$.
For the actual phase space sampling, we assume a Poisson distribution
and extract the mean number of gluon emissions in different rapidity
bins and fit the results to a linear function in $\Delta y_{j_f j_b}$,
finding a coefficient of $0.975$ for the inclusive production of a Higgs
boson with two jets. Here are the observed and fitted average gluon
multiplicities as a function of $\Delta y_{j_f j_b}$:
\begin{center}
\includegraphics[width=.75\textwidth]{ng_mean}
\end{center}
As shown for two rapidity slices the assumption of a Poisson
distribution is also a good approximation:
\begin{center}
\includegraphics[width=.49\textwidth]{{ng_1.5}.pdf}\hfill
\includegraphics[width=.49\textwidth]{{ng_5.5}.pdf}
\end{center}
\subsubsection{Number of Gluons inside Jets}
\label{sec:psp_ng_jet}
For each of the $n_g$ gluon emissions we can split the phase-space
integral into a (disconnected) region inside the jets and a remainder:
\begin{equation}
\label{eq:psp_split}
\int [\mathrm{d}p_i] = \int [\mathrm{d}p_i]\,
\theta\bigg(\sum_{j=1}^{m}{\cal J}_j(p_i)\bigg) + \int [\mathrm{d}p_i]\,
\bigg[1-\theta\bigg(\sum_{j=1}^{m}{\cal J}_j(p_i)\bigg)\bigg]\,.
\end{equation}
The next step is to decide how many of the gluons will form part of a
jet. This is done in the \lstinline!PhaseSpacePoint::sample_ng_jets!
function.
We choose an importance sampling which is flat in the plane
spanned by the azimuthal angle $\phi$ and the rapidity $y$. This is
observed in BFKL and valid in the limit of Multi-Regge-Kinematics
(MRK). Furthermore, we assume anti-$k_t$ jets, which cover an area of
$\pi R^2$.
In principle, the total accessible area in the $y$-$\phi$ plane is given
by $2\pi \Delta y_{fb}$, where $\Delta y_{fb}\geq \Delta y_{j_f j_b}$ is
the a priori unknown rapidity separation between the most forward and
backward partons. In most cases the extremal jets consist of single
partons, so that $\Delta y_{fb} = \Delta y_{j_f j_b}$. For the less common
case of two partons forming a jet we observe a maximum distance of $R$
between the constituents and the jet centre. In rare cases jets have
more than two constituents. Empirically, they are always within a
distance of $\tfrac{5}{3}R$ to the centre of the jet, so
$\Delta y_{fb} \leq \Delta y_{j_f j_b} + \tfrac{10}{3} R$. In practice, the
extremal partons are required to carry a large fraction of the jet
transverse momentum and will therefore be much closer to the jet axis.
In summary, for sufficiently large rapidity separations we can use the
approximation $\Delta y_{fb} \approx \Delta y_{j_f j_b}$. This scenario
is depicted here:
\begin{center}
\includegraphics[width=0.5\linewidth]{ps_large_y}
\end{center}
If there is no overlap between jets, the probability $p_{\cal J, >}$ for
an extra gluon to end up inside a jet is then given by
\begin{equation}
\label{eq:p_J_large}
p_{\cal J, >} = \frac{(m - 1)\*R^2}{2\Delta y_{j_f j_b}}\,.
\end{equation}
For a very small rapidity separation, eq.~\eqref{eq:p_J_large}
obviously overestimates the true probability. The maximum phase space
covered by jets in the limit of a vanishing rapidity distance between
all partons is $2mR \Delta y_{fb}$:
\begin{center}
\includegraphics[width=0.5\linewidth]{ps_small_y}
\end{center}
We therefore estimate the probability for a parton to end up inside a jet as
\begin{equation}
\label{eq:p_J}
p_{\cal J} = \min\bigg(\frac{(m - 1)\*R^2}{2\Delta y_{j_f j_b}}, \frac{mR}{\pi}\bigg)\,.
\end{equation}
Here we compare this estimate with the actually observed
fraction of additional emissions into jets as a function of the rapidity
separation:
\begin{center}
\includegraphics[width=0.75\linewidth]{pJ}
\end{center}
\subsubsection{Gluons outside Jets}
\label{sec:gluons_nonjet}
Using our estimate for the probability of a gluon to be a jet
constituent, we choose a number $n_{g,{\cal J}}$ of gluons inside
jets, which also fixes the number $n_g - n_{g,{\cal J}}$ of gluons
outside jets. As explained later on, we need to generate the momenta of
the gluons outside jets first. This is done in
\lstinline!PhaseSpacePoint::gen_non_jet!.
The azimuthal angle $\phi$ is generated flat within $0\leq \phi \leq 2
\pi$. The allowed rapidity interval is set by the most forward and
backward partons, which are necessarily inside jets. Since these parton
rapidities are not known at this point, we also have to postpone the
rapidity generation for the gluons outside jets. For the scalar
transverse momentum $p_\perp = |\mathbf{p}_\perp|$ of a gluon outside
jets we use the parametrisation
\begin{equation}
\label{eq:p_nonjet}
p_\perp = \lambda + \tilde{p}_\perp\*\tan(\tau\*r)\,, \qquad
\tau = \arctan\bigg(\frac{p_{\perp{\cal J}_\text{min}} - \lambda}{\tilde{p}_\perp}\bigg)\,.
\end{equation}
For $r \in [0,1)$, $p_\perp$ is always less than the minimum momentum
$p_{\perp{\cal J}_\text{min}}$ required for a jet. $\tilde{p}_\perp$ is
a free parameter, a good empirical value is $\tilde{p}_\perp = [1.3 +
0.2\*(n_g - n_{g,\cal J})]\,$GeV
\subsubsection{Resummation jet momenta}
\label{sec:ptj_res}
On the one hand, each jet momentum is given by the sum of its
constituent momenta. On the other hand, the resummation jet momenta are
fixed by the constraints in line five of the master
equation~\eqref{eq:resumdijetFKLmatched2}. We therefore have to
calculate the resummation jet momenta from these constraints before
generating the momenta of the gluons inside jets. This is done in
\lstinline!PhaseSpacePoint::reshuffle! and in the free
\lstinline!resummation_jet_momenta! function (declared in \texttt{resummation\_jet.hh}).
The resummation jet momenta are determined by the $\delta$ functions in
line five of eq.~(\ref{eq:resumdijetFKLmatched2}). The rapidities are
fixed to the rapidities of the jets in the input fixed-order events, so
that the FKL ordering is guaranteed to be preserved.
In traditional \HEJ reshuffling the transverse momentum are given through
\begin{equation}
\label{eq:ptreassign_old}
\mathbf{p}^B_{\mathcal{J}_{l\perp}} = \mathbf{j}_{l\perp} \equiv \mathbf{p}_{\mathcal{J}_{l}\perp}
+ \mathbf{q}_\perp \,\frac{|\mathbf{p}_{\mathcal{J}_{l}\perp}|}{P_\perp},
\end{equation}
where $\mathbf{q}_\perp = \sum_{j=1}^n \mathbf{p}_{i\perp}
\bigg[1-\theta\bigg(\sum_{j=1}^{m}{\cal J}_j(p_i)\bigg)\bigg] $ is the
total transverse momentum of all partons \emph{outside} jets and
$P_\perp = \sum_{j=1}^m |\mathbf{p}_{\mathcal{J}_{j}\perp}|$. Since the
total transverse momentum of an event vanishes, we can also use
$\mathbf{q}_\perp = - \sum_{j=1}^m
\mathbf{p}_{\mathcal{J}_{j}\perp}$. Eq.~(\ref{eq:ptreassign}) is a
non-linear system of equations in the resummation jet momenta
$\mathbf{p}_{\mathcal{J}_{l}\perp}$. Hence we would have to solve
\begin{equation}
\label{eq:ptreassign_eq}
\mathbf{p}_{\mathcal{J}_{l}\perp}=\mathbf{j}^B_{l\perp} \equiv\mathbf{j}_{l\perp}^{-1}
\left(\mathbf{p}^B_{\mathcal{J}_{l\perp}}\right)
\end{equation}
numerically.
Since solving such a system is computationally expensive, we instead
change the reshuffling around to be linear in the resummation jet
momenta. Hence~\eqref{eq:ptreassign_eq} gets replaces by
\begin{equation}
\label{eq:ptreassign}
\mathbf{p}_{\mathcal{J}_{l\perp}} = \mathbf{j}^B_{l\perp} \equiv \mathbf{p}^B_{\mathcal{J}_{l}\perp}
- \mathbf{q}_\perp \,\frac{|\mathbf{p}^B_{\mathcal{J}_{l}\perp}|}{P^B_\perp},
\end{equation}
which is linear in the resummation momentum. Consequently the equivalent
of~\eqref{eq:ptreassign_old} is non-linear in the Born momentum. However
the exact form of~\eqref{eq:ptreassign_old} is not relevant for the resummation.
Both methods have been tested for two and three jets with the \textsc{rivet}
standard analysis \texttt{MC\_JETS}. They didn't show any differences even
after $10^9$ events.
The reshuffling relation~\eqref{eq:ptreassign} allows the transverse
momenta $p^B_{\mathcal{J}_{l\perp}}$ of the fixed-order jets to be
somewhat below the minimum transverse momentum of resummation jets. It
is crucial that this difference does not become too large, as the
fixed-order cross section diverges for vanishing transverse momenta. In
the production of a Higgs boson with resummation jets above $30\,$GeV we observe
that the contribution from fixed-order events with jets softer than
about $20\,$GeV can be safely neglected. This is shown in the following
plot of the differential cross section over the transverse momentum of
the softest fixed-order jet:
\begin{center}
\includegraphics[width=.75\textwidth]{ptBMin}
\end{center}
Finally, we have to account for the fact that the reshuffling
relation~\eqref{eq:ptreassign} is non-linear in the Born momenta. To
arrive at the master formula~\eqref{eq:resumdijetFKLmatched2} for the
cross section, we have introduced unity in the form of an integral over
the Born momenta with $\delta$ functions in the integrand, that is
\begin{equation}
\label{eq:delta_intro}
1 = \int_{p_{j\perp}^B=0}^{p_{j\perp}^B=\infty}
\mathrm{d}^2\mathbf{p}_{j\perp}^B\delta^{(2)}(\mathbf{p}_{\mathcal{J}_{j\perp}}^B -
\mathbf{j}_{j\perp})\,.
\end{equation}
If the arguments of the $\delta$ functions are not linear in the Born
momenta, we have to compensate with additional Jacobians as
factors. Explicitly, for the reshuffling relation~\eqref{eq:ptreassign}
we have
\begin{equation}
\label{eq:delta_rewrite}
\prod_{l=1}^m \delta^{(2)}(\mathbf{p}_{\mathcal{J}_{l\perp}}^B -
\mathbf{j}_{l\perp}) = \Delta \prod_{l=1}^m \delta^{(2)}(\mathbf{p}_{\mathcal{J}_{l\perp}} -
\mathbf{j}_{l\perp}^B)\,,
\end{equation}
where $\mathbf{j}_{l\perp}^B$ is given by~\eqref{eq:ptreassign_eq} and only
depends on the Born momenta. We have extended the product to run to $m$
instead of $m-1$ by eliminating the last $\delta$ function
$\delta^{(2)}\!\!\left(\sum_{k=1}^n \mathbf{p}_{k\perp}\right )$.
The Jacobian $\Delta$ is the determinant of a $2m \times 2m$ matrix with $l, l' = 1,\dots,m$
and $X, X' = x,y$.
\begin{equation}
\label{eq:jacobian}
\Delta = \left|\frac{\partial\,\mathbf{j}^B_{l'\perp}}{\partial\, \mathbf{p}^B_{{\cal J}_l \perp}} \right|
= \left| \delta_{l l'} \delta_{X X'} - \frac{q_X\, p^B_{{\cal
J}_{l'}X'}}{\left|\mathbf{p}^B_{{\cal J}_{l'} \perp}\right| P^B_\perp}\left(\delta_{l l'}
- \frac{\left|\mathbf{p}^B_{{\cal J}_l \perp}\right|}{P^B_\perp}\right)\right|\,.
\end{equation}
The determinant is calculated in \lstinline!resummation_jet_weight!,
again coming from the \texttt{resummation\_jet.hh} header.
Having to introduce this Jacobian is not a disadvantage specific to the new
reshuffling. If we instead use the old reshuffling
relation~\eqref{eq:ptreassign_old} we \emph{also} have to introduce a
similar Jacobian since we actually want to integrate over the
resummation phase space and need to transform the argument of the
$\delta$ function to be linear in the resummation momenta for this.
\subsubsection{Gluons inside Jets}
\label{sec:gluons_jet}
After the steps outlined in section~\ref{sec:psp_ng_jet}, we have a
total number of $m + n_{g,{\cal J}}$ constituents. In
\lstinline!PhaseSpacePoint::distribute_jet_partons! we distribute them
randomly among the jets such that each jet has at least one
constituent. We then generate their momenta in
\lstinline!PhaseSpacePoint::split! using the \lstinline!Splitter! class.
The phase space integral for a jet ${\cal J}$ is given by
\begin{equation}
\label{eq:ps_jetparton} \prod_{i\text{ in }{\cal J}} \bigg(\int
\mathrm{d}\mathbf{p}_{i\perp}\ \int \mathrm{d} y_i
\bigg)\delta^{(2)}\Big(\sum_{i\text{ in }{\cal J}} \mathbf{p}_{i\perp} -
\mathbf{j}_{\perp}^B\Big)\delta(y_{\mathcal{J}}-y^B_{\mathcal{J}})\,.
\end{equation}
For jets with a single constituent, the parton momentum is obiously equal to the
jet momentum. In the case of two constituents, we observe that the
partons are always inside the jet cone with radius $R$ and often very
close to the jet centre. The following plots show the typical relative
distance $\Delta R/R$ for this scenario:
\begin{center}
\includegraphics[width=0.45\linewidth]{dR_2}
\includegraphics[width=0.45\linewidth]{dR_2_small}
\end{center}
According to this preference for small values of $\Delta R$, we
parametrise the $\Delta R$ integrals as
\begin{equation}
\label{eq:dR_sampling}
\frac{\Delta R}{R} =
\begin{cases}
0.25\,x_R & x_R < 0.4 \\
1.5\,x_R - 0.5 & x_R \geq 0.4
\end{cases}\,.
\end{equation}
Next, we generate $\Theta_1 \equiv \Theta$ and use the constraint $\Theta_2 = \Theta
\pm \pi$. The transverse momentum of the first parton is then given by
\begin{equation}
\label{eq:delta_constraints}
p_{1\perp} =
\frac{p_{\mathcal{J} y} - \tan(\phi_2) p_{\mathcal{J} x}}{\sin(\phi_1)
- \tan(\phi_2)\cos(\phi_1)}\,.
\end{equation}
We get $p_{2\perp}$ by exchanging $1 \leftrightarrow 2$ in the
indices. To obtain the Jacobian of the transformation, we start from the
single jet phase space eq.~(\ref{eq:ps_jetparton}) with the rapidity
delta function already rewritten to be linear in the rapidity of the
last parton, i.e.
\begin{equation}
\label{eq:jet_2p}
\prod_{i=1,2} \bigg(\int
\mathrm{d}\mathbf{p}_{i\perp}\ \int \mathrm{d} y_i
\bigg)\delta^{(2)}\Big(\mathbf{p}_{1\perp} + \mathbf{p}_{2\perp} -
\mathbf{j}_{\perp}^B\Big)\delta(y_2- \dots)\,.
\end{equation}
The integral over the second parton momentum is now trivial; we can just replace
the integral over $y_2$ with the equivalent constraint
\begin{equation}
\label{eq:R2}
\int \mathrm{d}R_2 \ \delta\bigg(R_2 - \bigg[\phi_{\cal J} - \arctan
\bigg(\frac{p_{{\cal J}y} - p_{1y}}{p_{{\cal J}x} -
p_{1x}}\bigg)\bigg]/\cos \Theta\bigg) \,.
\end{equation}
In order to fix the integral over $p_{1\perp}$ instead, we rewrite this
$\delta$ function. This introduces the Jacobian
\begin{equation}
\label{eq:jac_pt1}
\bigg|\frac{\partial p_{1\perp}}{\partial R_2} \bigg| =
\frac{\cos(\Theta)\mathbf{p}_{2\perp}^2}{p_{{\cal J}\perp}\sin(\phi_{\cal J}-\phi_1)}\,.
\end{equation}
The final form of the integral over the two parton momenta is then
\begin{equation}
\label{eq:ps_jet_2p}
\int \mathrm{d}R_1\ R_1 \int \mathrm{d}R_2 \int \mathrm{d}x_\Theta\ 2\pi \int
\mathrm{d}p_{1\perp}\ p_{1\perp} \int \mathrm{d}p_{2\perp}
\ \bigg|\frac{\partial p_{1\perp}}{\partial R_2} \bigg|\delta(p_{1\perp}
-\dots) \delta(p_{2\perp} - \dots)\,.
\end{equation}
As is evident from section~\ref{sec:psp_ng_jet}, jets with three or more
constituents are rare and an efficient phase-space sampling is less
important. For such jets, we exploit the observation that partons with a
distance larger than $R_{\text{max}} = \tfrac{5}{3} R$ to
the jet centre are never clustered into the jet. Assuming $N$
constituents, we generate all components
for the first $N-1$ partons and fix the remaining parton with the
$\delta$-functional. In order to end up inside the jet, we use the
parametrisation
\begin{align}
\label{eq:ps_jet_param}
\phi_i ={}& \phi_{\cal J} + \Delta \phi_i\,, & \Delta \phi_i ={}& \Delta
R_i
\cos(\Theta_i)\,, \\
y_i ={}& y_{\cal J} + \Delta y_i\,, & \Delta y_i ={}& \Delta
R_i
\sin(\Theta_i)\,,
\end{align}
and generate $\Theta_i$ and $\Delta R_i$ randomly with $\Delta R_i \leq
R_{\text{max}}$ and the empiric value $R_{\text{max}} = 5\*R/3$. We can
then write the phase space integral for a single parton as $(p_\perp = |\mathbf{p}_\perp|)$
\begin{equation}
\label{eq:ps_jetparton_x}
\int \mathrm{d}\mathbf{p}_{\perp}\ \int
\mathrm{d} y \approx \int_{\Box} \mathrm{d}x_{\perp}
\mathrm{d}x_{ R}
\mathrm{d}x_{\theta}\
2\*\pi\,\*R_{\text{max}}^2\,\*x_{R}\,\*p_{\perp}\,\*(p_{\perp,\text{max}}
- p_{\perp,\text{min}})
\end{equation}
with
\begin{align}
\label{eq:ps_jetparton_parameters}
\Delta \phi ={}& R_{\text{max}}\*x_{R}\*\cos(2\*\pi\*x_\theta)\,,&
\Delta y ={}& R_{\text{max}}\*x_{R}\*\sin(2\*\pi\*x_\theta)\,, \\
p_{\perp} ={}& (p_{\perp,\text{max}} - p_{\perp,\text{min}})\*x_\perp +
p_{\perp,\text{min}}\,.
\end{align}
$p_{\perp,\text{max}}$ is determined from the requirement that the total
contribution from the first $n-1$ partons --- i.e. the projection onto the
jet $p_{\perp}$ axis --- must never exceed the jet $p_\perp$. This gives
\todo{This bound is too high}
\begin{equation}
\label{eq:pt_max}
p_{i\perp,\text{max}} = \frac{p_{{\cal J}\perp} - \sum_{j<i} p_{j\perp}
\cos \Delta
\phi_j}{\cos \Delta
\phi_i}\,.
\end{equation}
The $x$ and $y$ components of the last parton follow immediately from
the first $\delta$ function. The last rapidity is fixed by the condition that
the jet rapidity is kept fixed by the reshuffling, i.e.
\begin{equation}
\label{eq:yJ_delta}
y^B_{\cal J} = y_{\cal J} = \frac 1 2 \ln \frac{\sum_{i=1}^n E_i+ p_{iz}}{\sum_{i=1}^n E_i - p_{iz}}\,.
\end{equation}
With $E_n \pm p_{nz} = p_{n\perp}\exp(\pm y_n)$ this can be rewritten to
\begin{equation}
\label{eq:yn_quad_eq}
\exp(2y_{\cal J}) = \frac{\sum_{i=1}^{n-1} E_i+ p_{iz}+p_{n\perp} \exp(y_n)}{\sum_{i=1}^{n-1} E_i - p_{iz}+p_{n\perp} \exp(-y_n)}\,,
\end{equation}
which is a quadratic equation in $\exp(y_n)$. The physical solution is
\begin{align}
\label{eq:yn}
y_n ={}& \log\Big(-b + \sqrt{b^2 + \exp(2y_{\cal J})}\,\Big)\,,\\
b ={}& \bigg(\sum_{i=1}^{n-1} E_i + p_{iz} - \exp(2y_{\cal J})
\sum_{i=1}^{n-1} E_i - p_{iz}\bigg)/(2 p_{n\perp})\,.
\end{align}
\todo{what's wrong with the following?} To eliminate the remaining rapidity
integral, we transform the $\delta$ function to be linear in the
rapidity $y$ of the last parton. The corresponding Jacobian is
\begin{equation}
\label{eq:jacobian_y}
\bigg|\frac{\partial y_{\cal J}}{\partial y_n}\bigg|^{-1} = 2 \bigg( \frac{E_n +
p_{nz}}{E_{\cal J} + p_{{\cal J}z}} + \frac{E_n - p_{nz}}{E_{\cal J} -
p_{{\cal J}z}}\bigg)^{-1}\,.
\end{equation}
Finally, we check that all designated constituents are actually
clustered into the considered jet.
\subsubsection{Final steps}
\label{sec:final}
Knowing the rapidity span covered by the extremal partons, we can now
generate the rapdities for the partons outside jets. We perform jet
clustering on all partons and check in
\lstinline!PhaseSpacePoint::jets_ok! that all the following criteria are
fulfilled:
\begin{itemize}
\item The number of resummation jets must match the number of
fixed-order jets.
\item No partons designated to be outside jets may end up inside jets.
\item All other outgoing partons \emph{must} end up inside jets.
\item The extremal (in rapidity) partons must be inside the extremal
jets. If there is, for example, an unordered forward emission, the
most forward parton must end up inside the most forward jet and the
next parton must end up inside second jet.
\item The rapidities of fixed-order and resummation jets must match.
\end{itemize}
After this, we adjust the phase-space normalisation according to the
third line of eq.~(\ref{eq:resumdijetFKLmatched2}), determine the
flavours of the outgoing partons, and adopt any additional colourless
bosons from the fixed-order input event. Finally, we use momentum
conservation to reconstruct the momenta of the incoming partons.
\subsection{The matrix element }
\label{sec:ME}
The derivation of the \HEJ matrix element is explained in some detail
in~\cite{Andersen:2017kfc}, where also results for leading and
subleading matrix elements for pure multijet production and production
of a Higgs boson with at least two associated jets are listed. Matrix
elements for $W$ and $Z/\gamma^*$ production together with jets are
given in~\cite{Andersen:2012gk,Andersen:2016vkp}, but not yet included.
The matrix elements are implemented in the \lstinline!MatrixElement!
class. To discuss the structure, let us consider the squared matrix
element for FKL multijet production with $n$ final-state partons:
\begin{align}
\label{eq:ME}
\begin{split}
\overline{\left|\mathcal{M}_\text{HEJ}^{f_1 f_2 \to f_1
g\cdots g f_2}\right|}^2 = \ &\frac {(4\pi\alpha_s)^n} {4\ (N_c^2-1)}
\cdot\ \textcolor{blue}{\frac {K_{f_1}(p_1^-, p_a^-)} {t_1}\ \cdot\ \frac{K_{f_2}(p_2^+, p_b^+)}{t_{n-1}}\ \cdot\ \left\|S_{f_1 f_2\to f_1 f_2}\right\|^2}\\
& \cdot \prod_{i=1}^{n-2} \textcolor{gray}{\left( \frac{-C_A}{t_it_{i+1}}\
V^\mu(q_i,q_{i+1})V_\mu(q_i,q_{i+1}) \right)}\\
& \cdot \prod_{j=1}^{n-1} \textcolor{red}{\exp\left[\omega^0(q_{j\perp})(y_{j+1}-y_j)\right]}.
\end{split}
\end{align}
The structure and momentum assignment of the unsquared matrix element is
as illustrated here:
\begin{center}
\includegraphics{HEJ_amplitude}
\end{center}
The square
of the complete matrix element as given in eq.~(\ref{eq:ME}) is
calculated by \lstinline!MatrixElement::operator()!. The last line of
eq.~\eqref{eq:ME} constitutes the all-order virtual correction,
-implemented in \lstinline!MatrixElement::virtual_corrections!. The
-remaining parts, which correspond to the square of the leading-order HEJ
-matrix element $\overline{\left|\mathcal{M}_\text{LO, HEJ}^{f_1f_2\to
-f_1g\cdots gf_2}\big(\big\{p^B_j\big\}\big)\right|}^{2}$, are computed
-in \lstinline!MatrixElement::tree!. We can further factor off the
+implemented in
+\lstinline!MatrixElement::virtual_corrections!.
+$\omega^0$ is the
+\textit{regularised Regge trajectory}
+\begin{equation}
+ \label{eq:omega_0}
+ \omega^0(q_\perp) = - C_A \frac{\alpha_s}{\pi} \log \left(\frac{q_\perp^2}{\lambda^2}\right)\,,
+\end{equation}
+where $\lambda$ is the slicing parameter limiting the softness of real
+gluon emissions, cf. eq.~(\ref{eq:resumdijetFKLmatched2}).
+
+The remaining parts, which correspond to the square of the leading-order
+HEJ matrix element $\overline{\left|\mathcal{M}_\text{LO,
+HEJ}^{f_1f_2\to f_1g\cdots
+gf_2}\big(\big\{p^B_j\big\}\big)\right|}^{2}$, are computed in
+\lstinline!MatrixElement::tree!. We can further factor off the
scale-dependent ``parametric'' part
\lstinline!MatrixElement::tree_param! containing all factors of the
strong coupling $4\pi\alpha_s$. Using this function saves some CPU time
when adjusting the renormalisation scale, see
section~\ref{sec:resum}. The remaining ``kinematic'' factors are
calculated in \lstinline!MatrixElement::kin!.
\subsubsection{FKL ladder and Lipatov vertices}
\label{sec:FKL_ladder}
The ``FKL ladder'' is the product
\begin{equation}
\label{eq:FKL_ladder}
\prod_{i=1}^{n-2} \left( \frac{-C_A}{t_it_{i+1}}\
V^\mu(q_i,q_{i+1})V_\mu(q_i,q_{i+1}) \right)
\end{equation}
appearing in the square of the matrix element for $n$ parton production,
cf. eq.~(\ref{eq:ME}), and implemented in
\lstinline!MatrixElement::FKL_ladder_weight!. The Lipatov vertex contraction
$V^\mu(q_i,q_{i+1})V_\mu(q_i,q_{i+1})$ is implemented \lstinline!C2Lipatovots!.
It is given by \todo{equation} \todo{mention difference between the two versions
of \lstinline!C2Lipatovots!, maybe even get rid of one}.
\subsubsection{Currents}
\label{sec:currents}
The current factors
$\frac{K_{f_1}K_{f_2}}{t_1 t_{n-1}}\left\|S_{f_1 f_2\to f_1
f_2}\right\|^2$ are implemented in the \lstinline!jM2!$\dots$ functions
of \texttt{src/currents.cc}. \todo{Only $\|S\|^2$ should be in currents}
\footnote{The current implementation for
Higgs production in \texttt{src/currents.cc} includes the $1/4$ factor
inside $S$, opposing to~\eqref{eq:ME}. Thus the overall normalisation is
unaffected.} The ``current acceleration multiplier'' (CAM) $K_{f}$
for a parton $f\in\{g,q,\bar{q}\}$ is defined as
\begin{align}
\label{eq:K_g}
K_g(p_1^-, p_a^-) ={}& \frac{1}{2}\left(\frac{p_1^-}{p_a^-} + \frac{p_a^-}{p_1^-}\right)\left(C_A -
\frac{1}{C_A}\right)+\frac{1}{C_A}\\
\label{eq:K_q}
K_q(p_1^-, p_a^-) ={}&K_{\bar{q}}(p_1^-, p_a^-) = C_F\,.
\end{align}
-The current contraction is given by\todo{check all this carefully!}
+The current contraction is given by\todo{check all this
+ carefully!}
\begin{equation}
\label{eq:S}
\left\|S_{f_1 f_2\to f_1 f_2}\right\|^2 = \sum_{\substack{\lambda_1 =
- +,-\\\lambda_2 = +,-}} \left|j^{\lambda_1}_\mu(p_1, p_a)\ j^{\lambda_2}_\mu(p_2, p_b)\right|^2\,.
+ +,-\\\lambda_2 = +,-}} \left|j^{\lambda_1}_\mu(p_1, p_a)\
+j^{\lambda_2}_\mu(p_2, p_b)\right|^2 = 2\sum_{\lambda =
+ +,-} \left|j^{-}_\mu(p_1, p_a)\ j^{\lambda}_\mu(p_2, p_b)\right|^2\,.
\end{equation}
The currents are independent of the parton flavour and read
\begin{equation}
\label{eq:j}
- j^\pm_\mu(p_{\text{out}}, p_{\text{in}}) = u^{\pm,\dagger}(p_{\text{out}})\ \sigma^\pm_\mu\ u^{\pm}(p_{\text{in}})\,,
+ j^\pm_\mu(p, q) = u^{\pm,\dagger}(p)\ \sigma^\pm_\mu\ u^{\pm}(q)\,,
\end{equation}
where $\sigma_\mu^\pm = (1, \pm \sigma_i)$ and $\sigma_i$ are the Pauli
matrices
\begin{equation}
\label{eq:Pauli_matrices}
\sigma_1 =
\begin{pmatrix}
0 & 1\\ 1 & 0
\end{pmatrix}
\,,
\qquad \sigma_2 =
\begin{pmatrix}
0 & -i\\ i & 0
\end{pmatrix}
\,,
\qquad \sigma_3 =
\begin{pmatrix}
1 & 0\\ 0 & -1
\end{pmatrix}
\,.
\end{equation}
The two-component chiral spinors are given by
\begin{align}
\label{eq:u_plus}
- u^+(p)={}& \left(\sqrt{p^+}, \sqrt{p^-} \frac{p_\perp}{|p_\perp|}\right) \,,\\
+ u^+(p)={}& \left(\sqrt{p^+}, \sqrt{p^-} \hat{p}_\perp \right) \,,\\
\label{eq:u_minus}
- u^-(p)={}& \left(\sqrt{p^-} \frac{p^*_\perp}{|p_\perp|}, -\sqrt{p^+}\right)\,,
+ u^-(p)={}& \left(\sqrt{p^-} \hat{p}^*_\perp, -\sqrt{p^+}\right)\,,
+\end{align}
+with $p^\pm = E\pm p_z,\, \hat{p}_\perp = \tfrac{p_\perp}{|p_\perp|},\,
+p_\perp = p_x + i p_y$. The spinors for vanishing transverse momentum
+are obtained by replacing $\hat{p_\perp} \to -1$.
+
+Explicitly, the currents read
+\begin{align}
+ \label{eq:j-_explicit}
+ j^-_\mu(p, q) ={}&
+ \begin{pmatrix}
+ \sqrt{p^+\,q^+} + \sqrt{p^-\,q^-} \hat{p}_{\perp} \hat{q}_{\perp}^*\\
+ \sqrt{p^-\,q^+}\, \hat{p}_{\perp} + \sqrt{p^+\,q^-}\,\hat{q}_{\perp}^*\\
+ -i \sqrt{p^-\,q^+}\, \hat{p}_{\perp} + i \sqrt{p^+\,q^-}\, \hat{q}_{\perp}^*\\
+ \sqrt{p^+\,q^+} - \sqrt{p^-\,q^-}\, \hat{p}_{\perp}\, \hat{q}_{\perp}^*
+ \end{pmatrix}\\
+ j^+_\mu(p, q) ={}&\big(j^-_\mu(p, q)\big)^*
+\end{align}
+If $q= p_{\text{in}}$ is the momentum of an incoming parton, we have
+$\hat{p}_{\text{in} \perp} = -1$ and either $p_{\text{in}}^+ = 0$ or
+$p_{\text{in}}^- = 0$. The current simplifies further:\todo{Helicities flipped w.r.t code}
+\begin{align}
+ \label{eq:j_explicit}
+ j^-_\mu(p_{\text{out}}, p_{\text{in}}) ={}&
+ \begin{pmatrix}
+ \sqrt{p_{\text{in}}^+\,p_{\text{out}}^+}\\
+ \sqrt{p_{\text{in}}^+\,p_{\text{out}}^-} \ \hat{p}_{\text{out}\,\perp}\\
+ -i\,j^-_1\\
+ j^-_0
+ \end{pmatrix}
+& p_{\text{in}\,z} > 0\,,\\
+ j^-_\mu(p_{\text{out}}, p_{\text{in}}) ={}&
+ \begin{pmatrix}
+ -\sqrt{p_{\text{in}}^-\,p_{\text{out}}^{-\phantom{+}}} \ \hat{p}_{\text{out}\,\perp}\\
+ - \sqrt{p_{\text{in}}^-\,p_{\text{out}}^+}\\
+ i\,j^-_1\\
+ -j^-_0
+ \end{pmatrix} & p_{\text{in}\,z} < 0\,.
\end{align}
-with $p^\pm = E\pm p_z,\, p_\perp = p_x + i p_y$. The spinors for
-vanishing transverse momentum are obtained by replacing
-$\frac{p_\perp}{|p_\perp|} \to -1$ (either $p^+$ or $p^-$ is zero in
-this limit). \textbf{Note: eqs.~(\ref{eq:j}) to (\ref{eq:u_minus})
- reproduce \lstinline!jio! for $p_{\text{in}\perp}=0$.}
\section{The fixed-order generator}
\label{sec:HEJFOG}
Even at leading order, standard fixed-order generators can only generate
events with a limited number of final-state particles within reasonable
CPU time. The purpose of the fixed-order generator is to supplement this
with high-multiplicity input events according to the first two lines of
eq.~\eqref{eq:resumdijetFKLmatched2} with the \HEJ approximation
$\mathcal{M}_\text{LO, HEJ}^{f_1f_2\to f_1g\cdots gf_2}$ instead of the
full fixed-order matrix element $\mathcal{M}_\text{LO}^{f_1f_2\to
f_1g\cdots gf_2}$. Its usage is described in the user
documentation \url{https://hej.web.cern.ch/HEJ/doc/current/user/HEJFOG.html}.
\subsection{File structure}
\label{sec:HEJFOG_structure}
The code for the fixed-order generator is in the \texttt{FixedOrderGen}
directory, which contains the following:
\begin{description}
\item[include:] Contains the C++ header files.
\item[src:] Contains the C++ source files.
\item[t:] Contains the source code for the automated tests.
\item[CMakeLists.txt:] Configuration file for the \cmake build system.
\item[configFO.yml:] Sample configuration file for the fixed-order generator.
\end{description}
The code is generally in the \lstinline!HEJFOG! namespace. Functions and
classes \lstinline!MyClass! are usually declared in
\texttt{include/MyClass.hh} and implemented in \texttt{src/MyClass.cc}.
\subsection{Program flow}
\label{sec:prog_flow}
A single run of the fixed-order generator consists of three or four
stages.
First, we perform initialisation similar to HEJ 2, see
section~\ref{sec:init}. Since there is a lot of overlap we frequently
reuse classes and functions from HEJ 2, i.e. from the
\lstinline!HEJ! namespace. The code for parsing the configuration file
is in \texttt{include/config.hh} and implemented in
\texttt{src/config.cc}.
If partial unweighting is requested in the user settings \url{https://hej.web.cern.ch/HEJ/doc/current/user/HEJFOG.html#settings},
the initialisation is followed by a calibration phase. We use a
\lstinline!EventGenerator! to produce a number of trial
events. We use these to calibrate the \lstinline!Unweighter! in
its constructor and produce a first batch of partially unweighted
events. This also allows us to estimate our unweighting efficiency.
In the next step, we continue to generate events and potentially
unweight them. Once the user-defined target number of events is reached,
we adjust their weights according to the number of required trials. As
in HEJ 2 (see section~\ref{sec:processing}), we pass the final
events to a \lstinline!HEJ::Analysis! and a
\lstinline!HEJ::CombinedEventWriter!.
\subsection{Event generation}
\label{sec:evgen}
Event generation is performed by the
\lstinline!EventGenerator::gen_event! member function. We begin by generating a
\lstinline!PhaseSpacePoint!. This is not to be confused with
the resummation phase space points represented by
\lstinline!HEJ::PhaseSpacePoint!! After jet clustering, we compute the
leading-order matrix element (see section~\ref{sec:ME}) and pdf factors.
The phase space point generation is performed in the
\lstinline!PhaseSpacePoint! constructor. We first construct the
user-defined number of $n_p$ partons (by default gluons) in
\lstinline!PhaseSpacePoint::gen_LO_partons!. We use flat sampling in
rapidity and azimuthal angle. For the scalar transverse momenta, we
distinguish between two cases. By default, they are generated based on a
random variable $x_{p_\perp}$ according to
\begin{equation}
\label{eq:pt_sampling}
p_\perp = p_{\perp,\text{min}} +
\begin{cases}
p_{\perp,\text{par}}
\tan\left(
x_{p_\perp}
\arctan\left(
\frac{p_{\perp,\text{max}} - p_{\perp,\text{min}}}{p_{\perp,\text{par}}}
\right)
\right)
& y < y_\text{cut}
\\
- \tilde{p}_{\perp,\text{par}}\log\left(1 - x_{p_\perp}\left[1 -
\exp\left(\frac{p_{\perp,\text{min}} -
p_{\perp,\text{max}}}{\tilde{p}_{\perp,\text{par}}}\right)\right]\right)
& y \geq y_\text{cut}
\end{cases}\,,
\end{equation}
where $p_{\perp,\text{min}}$ is the minimum jet transverse momentum,
$p_{\perp,\text{max}}$ is the maximum transverse parton momentum,
tentatively set to the beam energy, and $y_\text{cut}$, $p_{\perp,\text{par}}$
and $\tilde{p}_{\perp,\text{par}}$ are generation parameters set to
heuristically determined values of
\begin{align}
y_\text{cut}&=3,\\
p_{\perp,\text{par}}&=p_{\perp,\min}+\frac{n_p}{5}, \\
\tilde{p}_{\perp,\text{par}}&=\frac{p_{\perp,\text{par}}}{1 +
5(y-y_\text{cut})}.
\end{align}
The problem with this generation is that the transverse momenta peak at
the minimum transverse momentum required for fixed-order jets. However,
if we use the generated events as input for \HEJ resummation, events
with such soft transverse momenta hardly contribute, see
section~\ref{sec:ptj_res}. To generate efficient input for resummation,
there is the user option \texttt{peak pt}, which specifies the
dominant transverse momentum for resummation jets. If this option is
set, most jets will be generated as above, but with
$p_{\perp,\text{min}}$ set to the peak transverse momentum $p_{\perp,
\text{peak}}$. In addition, there is a small chance of around $2\%$ to
generate softer jets. The heuristic ansatz for the transverse momentum
distribution in the ``soft'' region is
\begin{equation}
\label{FO_pt_soft}
\frac{\partial \sigma}{\partial p_\perp} \propto e^{n_p\frac{p_\perp- p_{\perp,
\text{peak}}}{\bar{p}_\perp}}\,,
\end{equation}
where $n_p$ is the number of partons and $\bar{p}_\perp \approx
4\,$GeV. To achieve this distribution, we use
\begin{equation}
\label{eq:FO_pt_soft_sampling}
p_\perp = p_{\perp, \text{peak}} + \bar{p}_\perp \frac{\log x_{p_\perp}}{n_p}
\end{equation}
and discard the phase space point if the parton is too soft, i.e. below the threshold for
fixed-order jets.
After ensuring that all partons form separate jets, we generate any
potential colourless emissions. We then determine the incoming momenta
and flavours in \lstinline!PhaseSpacePoint::reconstruct_incoming! and
adjust the outgoing flavours to ensure an FKL configuration. Finally, we
may reassign outgoing flavours to generate suppressed (for example
unordered) configurations.
\subsection{Unweighting}
\label{sec:unweight}
Straightforward event generation tends to produce many events with small
weights. Those events have a negligible contribution to the final
observables, but can take up considerable storage space and CPU time in
later processing stages. This problem can be addressed by unweighting.
For naive unweighting, one would determine the maximum weight
$w_\text{max}$ of all events, discard each event with weight $w$ with a
probability $p=w/w_\text{max}$, and set the weights of all remaining
events to $w_\text{max}$. The downside to this procedure is that it also
eliminates a sizeable fraction of events with moderate weight, so that
the statistical convergence deteriorates.
To ameliorate this problem, we perform unweighting only for events with
sufficiently small weights. This is done by the
\lstinline!Unweighter! class. In the constructor we estimate the
mean and width of the weight-weight distribution from a sample of
events. We use these estimates to determine the maximum weight below
which unweighting is performed. The actual unweighting is the done in
the \lstinline!Unweighter::unweight! function.
\input{currents}
\bibliographystyle{JHEP}
\bibliography{biblio}
\end{document}