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diff --git a/doc/developer_manual/developer_manual.tex b/doc/developer_manual/developer_manual.tex
index 4392561..bf5f062 100644
--- a/doc/developer_manual/developer_manual.tex
+++ b/doc/developer_manual/developer_manual.tex
@@ -1,2175 +1,2176 @@
\documentclass[a4paper,11pt]{article}
\usepackage{fourier}
\usepackage[T1]{fontenc}
\usepackage{microtype}
\usepackage{geometry}
\usepackage{enumitem}
\setlist[description]{leftmargin=\parindent,labelindent=\parindent}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[utf8x]{inputenc}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{todonotes}
\usepackage{listings}
\usepackage{xspace}
\usepackage{tikz}
\usepackage{slashed}
\usepackage{subcaption}
\usetikzlibrary{arrows.meta}
\usetikzlibrary{shapes}
\usetikzlibrary{calc}
\usepackage[colorlinks,linkcolor={blue!50!black}]{hyperref}
\graphicspath{{build/figures/}{figures/}}
\usepackage[left]{showlabels}
\renewcommand{\showlabelfont}{\footnotesize\color{darkgreen}}
\emergencystretch \hsize
\setlength{\marginparwidth}{2cm}
\newcommand{\HEJ}{{\tt HEJ}\xspace}
\newcommand{\NLO}{{\tt NLO}\xspace}
\newcommand{\LO}{{\tt LO}\xspace}
\newcommand{\HIGHEJ}{\emph{High Energy Jets}\xspace}
\newcommand{\cmake}{\href{https://cmake.org/}{cmake}\xspace}
\newcommand{\html}{\href{https://www.w3.org/html/}{html}\xspace}
\newcommand{\YAML}{\href{http://yaml.org/}{YAML}\xspace}
\newcommand{\QCDloop}{\href{https://github.com/scarrazza/qcdloop}{QCDloop}\xspace}
\newcommand{\FORM}{{\tt FORM}\xspace}
\newcommand\matel[4][]{\mathinner{\langle#2\vert#3\vert#4\rangle}_{#1}}
\newcommand{\as}{\alpha_s}
\DeclareRobustCommand{\mathgraphics}[1]{\vcenter{\hbox{\includegraphics{#1}}}}
\def\spa#1.#2{\left\langle#1\,#2\right\rangle}
\def\spb#1.#2{\left[#1\,#2\right]} \def\spaa#1.#2.#3{\langle\mskip-1mu{#1} |
#2 | {#3}\mskip-1mu\rangle} \def\spbb#1.#2.#3{[\mskip-1mu{#1} | #2 |
{#3}\mskip-1mu]} \def\spab#1.#2.#3{\langle\mskip-1mu{#1} | #2 |
{#3}\mskip-1mu\rangle} \def\spba#1.#2.#3{\langle\mskip-1mu{#1}^+ | #2 |
{#3}^+\mskip-1mu\rangle} \def\spav#1.#2.#3{\|\mskip-1mu{#1} | #2 |
{#3}\mskip-1mu\|^2} \def\jc#1.#2.#3{j^{#1}_{#2#3}}
% expectation value
\newcommand{\ev}[1]{\text{E}\left[#1\right]}
\definecolor{darkgreen}{rgb}{0,0.4,0}
\lstset{ %
backgroundcolor=\color{lightgray}, % choose the background color; you must add \usepackage{color} or \usepackage{xcolor}
basicstyle=\footnotesize\usefont{T1}{DejaVuSansMono-TLF}{m}{n}, % the size of the fonts that are used for the code
breakatwhitespace=false, % sets if automatic breaks should only happen at whitespace
breaklines=false, % sets automatic line breaking
captionpos=t, % sets the caption-position to bottom
commentstyle=\color{red}, % comment style
deletekeywords={...}, % if you want to delete keywords from the given language
escapeinside={\%*}{*)}, % if you want to add LaTeX within your code
extendedchars=true, % lets you use non-ASCII characters; for 8-bits encodings only, does not work with UTF-8
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stepnumber=2, % the step between two line-numbers. If it's 1, each line will be numbered
stringstyle=\color{gray}, % string literal style
tabsize=2, % sets default tabsize to 2 spaces
title=\lstname,
emph={},
emphstyle=\color{darkgreen}
}
\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[README.md:] Basic information concerning \HEJ.
\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[examples:] Example analyses and scale setting code.
\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[current\_generator:] Contains the code for the current generator,
see section~\ref{sec:cur_gen}.
\item[FixedOrderGen:] Contains the code for the fixed-order generator,
see section~\ref{sec:HEJFOG}.
\item[COPYING:] License file.
\item[AUTHORS:] List of \HEJ authors.
\item[Changes-API.md:] Change history for the \HEJ API (application programming interface).
\item[Changes.md:] History of changes relevant for \HEJ users.
\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 sphinx
\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={}]
make doxygen
\end{lstlisting}
in your \texttt{build/} 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. Additionally each \lstinline!git push! is also
automatically tested via the GitLab CI (see appendix~\ref{sec:CI}).
\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. The code coverage of all
tests can be generated with \href{https://gcovr.com/en/stable/}{gcovr}.
Therefore add the flag \lstinline!-DTEST_COVERAGE=True! to cmake and run
\lstinline!make ctest_coverage!.
\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/bin/HEJ.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_analyses! function
creates a number objects 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_analyses! creates a number
of user-defined analyses loaded from one or more external libraries and
\textsc{rivet} analyses (see the user documentation
\url{https://hej.hepforge.org/doc/current/user/})
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!HEJ::EventReader! object for reading events
either in the the Les Houches event file format~\cite{Alwall:2006yp} or
an \href{https://www.hdfgroup.org/}{HDF5}-based
format~\cite{Hoeche:2019rti}. To allow making the decision at run time,
\lstinline!HEJ::EventReader! is an abstract base class defined in
\texttt{include/HEJ/EventReader.hh} and the implementations of the
derived classes are in \texttt{include/HEJ/LesHouchesReader.hh},
\texttt{include/HEJ/HDF5Reader.hh} and the corresponding \texttt{.cc}
source files in the \texttt{src} directory. The
\lstinline!LesHouchesReader! leverages
\href{http://home.thep.lu.se/~leif/LHEF/}{\texttt{include/LHEF/LHEF.h}}. 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}.
If unweighting is enabled, we also initialise an unweighter as defined
in \texttt{include/HEJ/Unweighter.hh}. The unweighting strategies are
explained in section~\ref{sec:unweight}.
\subsection{Event processing}
\label{sec:processing}
In the second stage events are continuously read from the event
file. After jet clustering, a number of corresponding resummation events
are generated for each input event and fed into the analyses 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!EventReader::read_event!\nodepart{second}{read event}};
\node
(data) [mynode,below=of reader]
{\lstinline!Event::EventData! constructor\nodepart{second}{convert to \HEJ object}};
\node
(cluster) [mynode,below=of data]
{\lstinline!Event::EventData::cluster!\nodepart{second}{cluster jets \&
classify \lstinline!EventType!}};
\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
(cut) [mynode,below=of resum]
{\lstinline!Analysis::pass_cuts!\nodepart{second}{apply cuts}};
\node
(unweight) [mynode,below=of cut]
{\lstinline!Unweighter::unweight!\nodepart{second}{unweight (optional)}};
\node
(fill) [mynode,below left=of unweight]
{\lstinline!Analysis::fill!\nodepart{second}{analyse event}};
\node
(write) [mynode,below right=of unweight]
{\lstinline!CombinedEventWriter::write!\nodepart{second}{write out event}};
\node
(control) [below=of unweight] {};
\draw[-{Latex[length=3mm, width=1.5mm]}]
(reader.south) -- node[left] {\lstinline!LHEF::HEPEUP!} (data.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(data.south) -- node[left] {\lstinline!Event::EventData!} (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]}]
($(resum.south)+(10mm, 0cm)$) -- ($(cut.north)+(10mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(resum.south)+(5mm, 0cm)$) -- ($(cut.north)+(5mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(resum.south)-(5mm, 0cm)$) -- ($(cut.north)-(5mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(resum.south)-(10mm, 0cm)$) -- node[left] {\lstinline!Event!} ($(cut.north)-(10mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(cut.south) -- (unweight.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cut.south)+(7mm, 0cm)$) -- ($(unweight.north)+(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cut.south)-(7mm, 0cm)$) -- node[left] {\lstinline!Event!} ($(unweight.north)-(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(unweight.south)-(3mm,0mm)$) .. controls ($(control)-(3mm,0mm)$) ..node[left] {\lstinline!Event!} (fill.east);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(unweight.south)-(3mm,0mm)$) .. controls ($(control)-(3mm,0mm)$) .. (write.west);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(unweight.south)+(3mm,0mm)$) .. controls ($(control)+(3mm,0mm)$) .. (fill.east);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(unweight.south)+(3mm,0mm)$) .. controls ($(control)+(3mm,0mm)$) ..node[right] {\lstinline!Event!} (write.west);
\end{tikzpicture}
\end{center}
\lstinline!EventData! is an intermediate container, its members are completely
accessible. In contrast after jet clustering and classification the phase space
inside \lstinline!Event! can not be changed any more
(\href{https://wikipedia.org/wiki/Builder_pattern}{Builder design pattern}). 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, the \lstinline!HDF5Writer! for
\href{https://www.hdfgroup.org/}{HDF5}~\cite{Hoeche:2019rti} and the
\lstinline!HepMC2Writer! or \lstinline!HepMC3Writer! 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-resummable, \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=1.5cm 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::EventData::cluster!\nodepart{second}{cluster jets}};
\node
(colour) [mynode,below=of cluster]
{\lstinline!Event::generate_colours()!\nodepart{second}{generate particle colour}};
\node
(gen_scales) [mynode,below=of colour]
{\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) -- (colour.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cluster.south)+(7mm, 0cm)$) -- ($(colour.north)+(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(cluster.south)-(7mm, 0cm)$) -- node[left]
{\lstinline!Event!} ($(colour.north)-(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
(colour.south) -- (gen_scales.north);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(colour.south)+(7mm, 0cm)$) -- ($(gen_scales.north)+(7mm, 0cm)$);
\draw[-{Latex[length=3mm, width=1.5mm]}]
($(colour.south)-(7mm, 0cm)$) -- 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}
\subsubsection{HEJ@NLO Truncation}
\label{sec:hejnlo}
For predictions involving NLO matching, we need to multiply our inclusive HEJ prediction, $\sigma^{2j}_{\HEJ}$, by an appropriate reweighting factor,
\begin{equation}
\sigma^{2j}_{\HEJ} \left(\frac{\sigma^{2j}_{\NLO}}{\sigma^{2j}_{\tt HEJ@NLO}}\right) = \sigma^{2j}_{\HEJ} \left(\frac{\sigma^{2j}_{\NLO}}{\sigma^{2jE}_{\tt HEJ@NLO}+ \sigma^{3j}_{\LO},}\right)
\end{equation}
with an analogous equation for a histogram bin. To obtain the exclusive $\sigma^{2jE}_{\tt HEJ@NLO}$ cross section, we can enable the \lstinline!NLO truncation! options in the config file at runtime. This option restricts the method \lstinline!PhaseSpacePoint::sample_ng! to only generate 0 or 1 additional gluons for the real emissions and then truncates to the first term in $\Delta y$ in the \lstinline!MatrixElement! virtual corrections for the case of 0 extra emissions.
\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 $x_b$ of the hadronic centre-of-mass energy $\sqrt{s}$,
\begin{align}
\label{eq:x_a}
x_a ={}& \frac{1}{\sqrt{s}}\sum_{i=1}^{n}(E_i - p_{iz}),\\
\label{eq:x_b}
x_b ={}& \frac{1}{\sqrt{s}}\sum_{i=1}^{n}(E_i + p_{iz}),
\end{align}
The square of the \emph{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.hepforge.org/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}
For configurations beyond leading-log, gluon emission is not allowed
everywhere and the rapidity span has to be adjusted. When determining
the most forward and backward jets, we exclude unordered emissions and
the extremal jet in extremal quark-antiquark emissions. In addition,
we subtract the rapidity region between central quark and antiquark
emission. Technically, we could have two or more additional gluons
inside one of the latter jets, in which case there could be gluons
that end up in between the jet centres but still outside the region
spanned by quark and antiquark. However, the phase space for this is
very small and should not affect the average number of emitted gluons
very much.
\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}
Again, for configurations beyond leading-log the allowed rapidity
range has to be modified as explained in section~\ref{sec:psp_ng}. We
also subtract the areas of unordered jets and the extremal jet
in extremal quark-antiquark emission when we estimate the phase space
area covered by jets. The reason is that in this case any additional
emission inside such a jet would have to beyond the hard parton
dominating the next jet in rapidity, which is possible but rare.
\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}
Since \HEJ generated additional soft emissions, their recoil have to be
compensated through the remaining particles. In the traditional reshuffling this
recoil was only distributed over all jets. However in anticipation of including
more bosons ($WW+2j$), or less jets ($H+1j$) this changed to also include bosons
in the reshuffling. In the following we will still always call the recoilers
\emph{jet} instead of generic \emph{Born particles}, since the bosons are
treated completely equivalent to (massive) jets.
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{Colour connection}
\label{sec:Colour}
\begin{figure}
\input{src/ColourConnect.tex}
\caption{Left: Non-crossing colour flow dominating in the MRK limit. The
crossing of the colour line connecting to particle 2 can be resolved by
writing particle 2 on the left. Right: A colour flow with a (manifest)
colour-crossing. The crossing can only be resolved if one breaks the
rapidities order, e.g. switching particles 2 and 3. From~\cite{Andersen:2017sht}.}
\label{fig:Colour_crossing}
\end{figure}
After the phase space for the resummation event is generated, we can construct
the colour for each particle. To generate the colour flow one has to call
\lstinline!Event::generate_colours! on any \HEJ configuration. For non-\HEJ
event we do not change the colour, and assume it is provided by the user (e.g.
through the LHE file input). The colour connection is done in the large $N_c$
(infinite number of colour) limit with leading colour in
MRK~\cite{Andersen:2008ue, Andersen:2017sht}. The idea is to allow only
$t$-channel colour exchange, without any crossing colour lines. For example the
colour crossing in the colour connection on the left of
figure~\ref{fig:Colour_crossing} can be resolved by switching \textit{particle
2} to the left.
We can write down the colour connections by following the colour flow from
\textit{gluon a} to \textit{gluon b} and back to \textit{gluon a}, e.g.
figure~\ref{fig:Colour_a123ba} corresponds to $a123ba$. All valid, non-crossing
colour flow will connect all external legs, while respecting the rapidity
ordering. For example the left of figure~\ref{fig:Colour_crossing} is allowed
($a134b2a$), it can be dientangled similar to figure~\ref{fig:Colour_a13b2a}.
However the right of the same figures breaks the rapidity ordering between 2 and
3 ($a1324ba$). Connections between $b$ and $a$ are in inverse order, i.e.
$ab321a$ corresponds to~\ref{fig:Colour_a123ba} ($a123ba$) with colour and
anti-colour swapped.
\begin{figure}
\centering
\subcaptionbox{$a123ba$\label{fig:Colour_a123ba}}{
\includegraphics[height=0.25\textwidth]{colour_a123ba.pdf}}
\subcaptionbox{$a13b2a$\label{fig:Colour_a13b2a}}{
\includegraphics[height=0.25\textwidth]{colour_a13b2a.pdf}}
\subcaptionbox{$a\_123ba$\label{fig:Colour_a_123ba}}{
\includegraphics[height=0.25\textwidth]{colour_a_123ba.pdf}}
\subcaptionbox{$a13b2\_a$\label{fig:Colour_a13b2_a}}{
\includegraphics[height=0.25\textwidth]{colour_a13b2_a.pdf}}
\subcaptionbox{$a14b3\_2a$\label{fig:Colour_a14b3_2a}}{
\includegraphics[height=0.25\textwidth]{colour_a14b3_2a.pdf}}
\caption{Different colour non-crossing colour connections. Both incoming
particles are drawn at the top or bottom and the outgoing left or right.
Outside arrows represent the the colour flow.}
\end{figure}
If we replace two gluons with a quark, (anti-)quark pair we break one of the
colour connections. We denote such a connection by a underscore (e.g. $1\_a$).
For example the equivalent of~\ref{fig:Colour_a123ba} ($a123ba$) with an
incoming antiquark is~\ref{fig:Colour_a_123ba} ($a\_123ba$). Such colour flows
are always possible for all leading and next-to-leading configurations, like
unordered emission~\ref{fig:Colour_a13b2_a} or central
$q\bar{q}$-pairs~\ref{fig:Colour_a14b3_2a} \footnote{Some sub-subleading
configurations are also possible. We could trivially chain multiple subleading
blocks, like $qQ\to gqQg$ from two unordered currents.}. We treat subleading
connections as one blob in which both $t$- and $u$-channels colour exchange is
allowed. E.g. $u$-channel equivalent of~\ref{fig:Colour_a14b3_2a} is
$a13\_24ba$. Each subleading blobs connect like gluons to the colour lines.
Some rapidity ordering allow multiple colour connections,
e.g.~\ref{fig:Colour_a123ba} and~\ref{fig:Colour_a13b2a}. This is always the
case if a gluon (or subleading blob) radiates off a gluon line. In that case we
randomly connect the gluon (or subleading blob) to either the colour or
anti-colour. In the generation we keep track whether we are on a quark or gluon
line, and act accordingly.
\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 $Z/\gamma^*$ production together with jets are
given in~\cite{Andersen:2016vkp}.
A full list of all implemented currents is given in
section~\ref{sec:currents_impl}.
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}_\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_n^+, 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.~\eqref{eq:ME} is
calculated by \lstinline!MatrixElement::operator()!. The \textcolor{red}{last line} of
eq.~\eqref{eq:ME} constitutes the all-order virtual correction,
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.~\eqref{eq:resumdijetFKLmatched2}. $\lambda$ can be
changed at runtime by setting \lstinline!regulator parameter! in
the configuration file.
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!.
The ``colour acceleration multiplier'' (CAM) $\textcolor{blue}{K_{f}}$
for a parton $f\in\{g,q,\bar{q}\}$ is defined as
\begin{align}
\label{eq:K_g}
\textcolor{blue}{ 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}
\textcolor{blue}{K_q(p_1^-, p_a^-)} ={}&\textcolor{blue}{K_{\bar{q}}(p_1^-, p_a^-)} = C_F\,.
\end{align}
and the helicity-summed current contraction squares $\left\|S_{f_1
f_2\to f_1 f_2}\right\|^2$ are explained in
section~\ref{sec:currents}.
\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{Processes with interference}
\label{sec:interference}
For some processes several contributions can produce the same final state and it is necessary to take into account the interference between them. As an example let us consider the FKL squared matrix element for $Z/\gamma^*$ emission with two incoming quarks. It reads~\cite{Andersen:2016vkp}
\begin{align}
\label{eq:ME_Z}
\begin{split}
\left|\mathcal{M}_{\HEJ}^{qQ\to Z qg\cdots gQ}\right|^2 &=\ \frac{g_s^4( g_s^2
C_A)^{n-2}}{4(N_c^2-1)}\cdot\textcolor{blue}{K_q(p_1^-,p_a^-)\cdot K_q(p_n^+,p_b^+)}\\
\times \Bigg(& \textcolor{blue}{\frac{\| j_a^{Z}\cdot
j_b\|^2}{t_{a1}t_{a(n-1)}}}
\prod^{n-2}_{i=1} \textcolor{gray}{\frac{-V^2(q_{ai},
q_{a(i+1)})}{t_{ai} t_{a(i+1)}}} \prod_{i=1}^{n-1}
\textcolor{red}{\exp(\omega^0(q_{ai\perp}) (y_{i+1} - y_i))}\\
+\ &\textcolor{blue}{\frac{\|j_a \cdot j_b^{Z} \|^2}{t_{b1}t_{b(n-1)}}}
\prod^{n-2}_{i=1}\textcolor{gray}{\frac{-V^2(q_{bi}, q_{b(i+1)})}{t_{bi} t_{b(i+1)}}} \prod_{i=1}^{n-1} \textcolor{red}{\exp(\omega^0(q_{bi\perp})(y_{i+1} - y_i))} \\
-\ &\textcolor{blue}{\frac{2\Re\{ (j_a^{Z}\cdot j_b)(\overline{j_a \cdot
j_b^{Z}})\}}{\sqrt{t_{a1}t_{b1}}\sqrt{t_{a(n-1)} t_{b(n-1)}}}}\\
& \; \times \prod^{n-2}_{i=1}\textcolor{gray}{\frac{V(q_{ai}, q_{a(i+1)})\cdot V(q_{bi},
q_{b(i+1)})}{\sqrt{t_{ai}t_{bi}} \sqrt{t_{a(i+1)}t_{b(i+1)}}}} \prod_{i=1}^{n-1} \textcolor{red}{\exp(\omega^0(\sqrt{q_{ai\perp}q_{bi\perp}})(y_{i+1} - y_i))}\Bigg),
\end{split}
\end{align}
where a sum over helicities in the products of currents is
implied. The first two terms correspond to a boson emission from
either the upper quark line (labeled $a$) or lower quark line (labeled
$b$) in both the amplitude and the complex conjugate, while the last
term is the interference between the top and bottom emissions. To deal
with this structure the intermediate objects are kept as vectors. For
example, \lstinline!MatrixElement::FKL_ladder_weight_mix! returns the
ladder factors for top, bottom and mixed emission, the latter one
making use of \lstinline!C2Lipatovots_Mix! which implements the
contraction
$V^\mu(q_{ai},q_{a(i+1)})V_\mu(q_{bi},q_{b(i+1)})$. \lstinline!MatrixElement::operator()!
and \lstinline!MatrixElement::tree! then use these vectors to perform
the sum in eq.~(\ref{eq:ME_Z}).
\subsubsection{Matrix elements for Higgs plus jets}
\label{sec:ME_h_jets}
In the production of a Higgs boson together with jets the parametric
parts and the virtual corrections only require minor changes in the
respective functions. However, in the ``kinematic'' parts we have to
distinguish between several cases, which is done in
\lstinline!MatrixElement::tree_kin_Higgs!. The Higgs boson can be
\emph{central}, i.e. inside the rapidity range spanned by the extremal
partons (\lstinline!MatrixElement::tree_kin_Higgs_central!) or
\emph{peripheral} and outside this range
(\lstinline!MatrixElement::tree_kin_Higgs_first! or
\lstinline!MatrixElement::tree_kin_Higgs_last!).
If a Higgs boson with momentum $p_H$ is emitted centrally, after parton
$j$ in rapidity, the matrix element reads
\begin{equation}
\label{eq:ME_h_jets_central}
\begin{split}
\overline{\left|\mathcal{M}_\HEJ^{f_1 f_2 \to f_1 g\cdot H
\cdot g f_2}\right|}^2 = \ &\frac {\alpha_s^2 (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{1}{t_j t_{j+1}} \cdot\ \frac{K_{f_2}(p_n^+, p_b^+)}{t_{n}}\ \cdot\ \left\|S_{f_1
f_2\to f_1 H f_2}\right\|^2}\\
& \cdot \prod_{\substack{i=1\\i \neq j}}^{n-1} \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 \textcolor{red}{\prod_{i=1}^{n-1}
\exp\left[\omega^0(q_{i\perp})\Delta y_i\right]}
\end{split}
\end{equation}
with the momentum definitions
\begin{center}
\includegraphics{HEJ_central_Higgs_amplitude}
\end{center}
$q_i$ is the $i$th $t$-channel momentum and $\Delta y_i$ the rapidity
gap between outgoing \emph{particles} (not partons) $i$ and $i+1$ in
rapidity ordering.
For \emph{peripheral} emission in the backward direction
(\lstinline!MatrixElement::tree_kin_Higgs_first!) we first check whether
the most backward parton is a gluon or an (anti-)quark. In the latter
case the leading contribution to the matrix element arises through
emission off the $t$-channel gluons and we can use the same formula
eq.~(\ref{eq:ME_h_jets_central}) as for central emission. If the most
backward parton is a gluon, the square of the matrix element can be
written as
\begin{equation}
\label{eq:ME_h_jets_peripheral}
\begin{split}
\overline{\left|\mathcal{M}_\HEJ^{g f_2 \to H g\cdot g f_2}\right|}^2 = \ &\frac {\alpha_s^2 (4\pi\alpha_s)^n} {4(N_c^2-1)}
\textcolor{blue}{\cdot\ \frac{1}{t_1}\
\frac{K_{f_2}(p_n^+, p_b^+)}{t_{n-1}}\ \cdot\ \left\|S_{g
f_2\to H f_2}\right\|^2}\\
& \cdot \prod_{\substack{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 \textcolor{red}{\prod_{i=1}^{n-1}
\exp\left[\omega^0(q_{i\perp}) (y_{i+1} - y_i)\right]}\,.
\end{split}
\end{equation}
The current contraction $\textcolor{blue}{S_{gf_2\to H f_2}}$ has only
one final-state parton; in contrast to all other \HEJ processes it is
enough to have a \emph{single} jet together with the Higgs boson.
The momenta are assigned as follows:
\begin{center}
\includegraphics{HEJ_peripheral_Higgs_amplitude}
\end{center}
The \textcolor{blue}{blue part} is implemented in
\lstinline!MatrixElement::MH2_backwardH!. The actual current contraction $\textcolor{blue}{\|S\|^2}$
is calculated in \lstinline!ME_jgH_j! inside
\lstinline!src/Hjets.cc!. All other building blocks are
already available.
The forward emission of a Higgs boson is completely analogous. We can
use the same function \lstinline!MatrixElement::MH2_backwardH!, replacing
$p_n \to p_1 ,\, p_b \to p_a$.
\subsubsection{Currents}
\label{sec:currents}
The current factors $\frac{K_{f_1}K_{f_2}}{t_1 t_{n-1}}\left\|S_{f_1
f_2\to f_1 f_2}\right\|^2$ and their extensions for unordered and
quark-antiquark emissions are implemented in the
\lstinline!ME_!$\dots$ functions in \texttt{src/MatrixElement.cc}. The
current contractions $\left\|S_{\dots}\right\|^2$ are unfortunately
also called \lstinline!ME_!$\dots$ and are implemented in
\texttt{src/jets.cc}.
The corresponding functions with additional boson emission are currently in
\texttt{src/Hjets.cc}, \texttt{src/Wjets.cc}, \texttt{src/WWjets.cc},
and \texttt{src/Zjets.cc}. \todo{Only $\|S\|^2$ should be in currents.}
Formulas for the current contractions are listed in section~\ref{sec:currents_impl}.
\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. Naive unweighting can be
performed by using the \lstinline!set_cut_to_maxwt! member function of the
\lstinline!Unweighter! on the events and then call the
\lstinline!unweight! member function. It can be enabled for the
resummation events as explained in the user documentation.
To ameliorate the problem of naive unweighting, we also implement
partial unweighting. That is, we perform unweighting only for events
with sufficiently small weights. When using the \lstinline!Unweighter!
member function \lstinline!set_cut_to_peakwt! 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; events with a larger weight are not
touched. The actual unweighting is again done in the
\lstinline!Unweighter::unweight! function.
To estimate the peak weight we employ the following heuristic
algorithm. For a calibration sample of $n$ events, create a histogram
with $b=\sqrt{n}$ equal-sized bins. The histogram ranges from $
\log(\min |w_i|)$ to $\log(|\max w_i|)$, where $w_i$ are the event weights. For
each event, add $|w_i|$ to the corresponding bin. We then prune the
histogram by setting all bins containing less than $c=n/b$
events to zero. This effectively removes statistical outliers. The
logarithm of the peak weight is then the centre of the highest bin in
the histogram. In principle, the number of bins $b$ and the pruning
parameter $c$ could be tuned further.
To illustrate the principle, here is a weight-weight histogram
filled with a sample of 100000 event weights before the pruning:
\begin{center}
\includegraphics[width=0.7\linewidth]{wtwt}
\end{center}
The peaks to the right are clearly outliers caused by single
events. After pruning we get the following histogram:
\begin{center}
\includegraphics[width=0.7\linewidth]{wtwt_cut}
\end{center}
The actual peak weight probably lies above the cut, and the algorithm
can certainly be improved. Still, the estimate we get from the pruned
histogram is already good enough to eliminate about $99\%$ of the
low-weight events.
\section{The current generator}
\label{sec:cur_gen}
The current generator in the \texttt{current\_generator} directory is
automatically invoked when building \HEJ 2. Its task is to compute and
generate C++ code for the current contractions listed in
section~\ref{sec:currents}. For each source file \texttt{<j\_j>.frm}
inside \texttt{current\_generator} it generates a corresponding header
\texttt{include/HEJ/currents/<j\_j>.hh} inside the build directory. The
header can be included with
\begin{lstlisting}[language=C++,caption={}]
#include "HEJ/currents/<j>.hh"
\end{lstlisting}
The naming scheme is
\begin{itemize}
\item \texttt{j1\_j2.frm} for the contraction of current \texttt{j1} with current \texttt{j2}.
\item \texttt{j1\_vx\_j2.frm} for the contraction of current \texttt{j1}
with the vertex \texttt{vx} and the current \texttt{j2}.
\end{itemize}
For instance, \texttt{juno\_qqbarW\_j.frm} would indicate the
contraction of an unordered current with a (central) quark-antiquark-W
emission vertex and a standard FKL current.
\subsection{Implementing new current contractions}
\label{sec:cur_gen_new}
Before adding a new current contraction, first find a representation
where all momenta that appear inside the currents are \textcolor{red}{lightlike}
and have \textcolor{red}{positive energy}. Section~\ref{sec:p_massive} describes
how massive momenta can be decomposed into massless ones. For momenta
$p'$ with negative energies replace $p' \to - p$ and exploit that
negative- and positive-energy spinors are related by a phase, which is
usually irrelevant: $u^\lambda(-p) = \pm i u^\lambda(p)$.
To implement a current contraction \lstinline!jcontr! create a new
file \texttt{jcontr.frm} inside the \texttt{current\_generator}
directory. \texttt{jcontr.frm} should contain \FORM code, see
\url{https://www.nikhef.nl/~form/} for more information on
\FORM. Here is a small example:
\begin{lstlisting}[caption={}]
* FORM comments are started by an asterisk * at the beginning of a line
* First include the relevant headers
#include- include/helspin.frm
#include- include/write.frm
* Define the symbols that appear.
* UPPERCASE symbols are reserved for internal use
vectors p1,...,p10;
indices mu1,...,mu10;
* Define local expressions of the form [NAME HELICITIES]
* for the current contractions for all relevant helicity configurations
#do HELICITY1={+,-}
#do HELICITY2={+,-}
* We use the Current function
* Current(h, p1, mu1, ..., muX, p2) =
* u(p1) \gamma_{mu1} ... \gamma_{muX} u(p2)
* where h=+1 or h=-1 is the spinor helicity.
* All momenta appearing as arguments have to be *lightlike*
local [jcontr `HELICITY1'`HELICITY2'] =
Current(`HELICITY1'1, p1, mu1, p2, mu2, p3)
*Current(`HELICITY2'1, p4, mu2, p2, mu1, p1);
#enddo
#enddo
.sort
* Main procedure that calculates the contraction
#call ContractCurrents
.sort
* Optimise expression
format O4;
* Format in a (mostly) c compatible way
format c;
* Write start of C++ header file
#call WriteHeader(`OUTPUT')
* Write a template function jcontr
* taking as template arguments two helicities
* and as arguments the momenta p1,...,p4
* returning the contractions [jcontr HELICITIES] defined above
#call WriteOptimised(`OUTPUT',jcontr,2,p1,p2,p3,p4)
* Wrap up
#call WriteFooter(`OUTPUT')
.end
\end{lstlisting}
\subsection{Calculation of contractions}
\label{sec:contr_calc}
In order to describe the algorithm for the calculation of current
contractions we first have to define the currents and establish some
useful relations.
\subsubsection{Massive momenta}
\label{sec:p_massive}
We want to use relations for lightlike momenta. Momenta $P$ that are
\emph{not} lightlike can be written as the sum of two lightlike
momenta:
\begin{equation}
\label{eq:P_massive}
P^\mu = p^\mu + q^\mu\,, \qquad p^2 = q^2 = 0 \,.
\end{equation}
This decomposition is not unique. If we impose the arbitrary
constraint $q_\perp = 0$ and require real-valued momentum
components we can use the ansatz
\begin{align}
\label{eq:P_massive_p}
p_\mu ={}& P_\perp\*(\cosh y, \cos \phi, \sin \phi, \sinh y)\,,\\
\label{eq:P_massive_q}
q_\mu ={}& (E, 0, 0, s\,E)\,,
\end{align}
where $P_\perp$ is the transverse momentum of $P$ and $\phi$ the corresponding azimuthal angle. For the remaining parameters we obtain
\begin{align}
\label{eq:P_massive_plus}
P^+ > 0:& & y ={}& \log \frac{P^+}{P_\perp}\,,\qquad E = \frac{P^2}{2P^+}\,,\qquad s = -1\,,\\
\label{eq:P_massive_minus}
P^- > 0:& & y ={}& \log \frac{P_\perp}{P^-}\,,\qquad E = \frac{P^2}{2P^-}\,,\qquad s = +1\,.
\end{align}
The decomposition is implemented in the \lstinline!split_into_lightlike! function.
\subsubsection{Currents and current relations}
\label{sec:current_relations}
Our starting point are generalised currents
\begin{equation}
\label{eq:j_gen}
j^{\pm}(p, \mu_1,\dots,\mu_{2N-1},q) = \bar{u}^{\pm}(p)\gamma_{\mu_1} \dots \gamma_{\mu_{2N-1}} u^\pm(q)\,.
\end{equation}
Since there are no masses, we can consider two-component chiral spinors
\begin{align}
\label{eq:u_plus}
u^+(p)={}& \left(\sqrt{p^+}, \sqrt{p^-} \hat{p}_\perp \right) \,,\\
\label{eq:u_minus}
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 $\hat{p}_\perp \to -1$.
This gives
\begin{equation}
\label{eq:jminus_gen}
j^-(p,\mu_1,\dots,\mu_{2N-1},q) = u^{-,\dagger}(p)\ \sigma^-_{\mu_1}\ \sigma^+_{\mu_2}\dots\sigma^-_{\mu_{2N-1}}\ u^{-}(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}
For positive-helicity currents we can either flip all helicities in
eq.~(\ref{eq:jminus_gen}) or reverse the order of the arguments, i.e.
\begin{equation}
\label{eq:jplus_gen}
j^+(p,\mu_1,\dots,\mu_{2N-1},q) = \big(j^-(p,\mu_1,\dots,\mu_{2N-1},q)\big)^* = j^-(q,\mu_{2N-1},\dots,\mu_1,p) \,.
\end{equation}
Using the standard spinor-helicity notation we have
\begin{gather}
\label{eq:spinors_spinhel}
u^+(p) = | p \rangle\,, \qquad u^-(p) = | p ]\,, \qquad u^{+,\dagger}(p) = [ p |\,, \qquad u^{-,\dagger}(p) = \langle p |\,,\\
\label{eq:current_spinhel}
j^-(p,\mu_1,\dots,\mu_{2N-1},q) = \langle p |\ \mu_1\ \dots\ \mu_{2N-1}\ | q ] \,.\\
\label{eq:contraction_spinhel}
P_{\mu_i} j^-(p,\mu_1,\dots,\mu_{2N-1},q) = \langle p |\ \mu_1\ \dots\ \mu_{i-1}\ P\ \mu_{i+1}\ \dots\ \mu_{2N-1}\ | q ] \,.
\end{gather}
Lightlike momenta $p$ can be decomposed into spinor products:
\begin{equation}
\label{eq:p_decomp}
\slashed{p} = |p\rangle [p| + |p] \langle p |\,.
\end{equation}
Taking into account helicity conservation this gives the following relations:
\begingroup
\addtolength{\jot}{1em}
\begin{align}
\label{eq:p_in_current}
\langle p |\ \mu_1\ \dots\ \mu_i\ P\ \mu_{i+1}\ \dots\ \mu_{2N-1}\ | q ] ={}&
\begin{cases}
\langle p |\ \mu_1\ \dots\ \mu_i\ |P]\ \langle P|\ \mu_{i+1}\ \dots\ \mu_{2N-1}\ | q ]& i \text{ even}\\
\langle p |\ \mu_1\ \dots\ \mu_i\ |P\rangle\ [ P|\ \mu_{i+1}\ \dots\ \mu_{2N-1}\ | q ]& i \text{ odd}
\end{cases}\,,\\
\label{eq:p_in_angle}
\langle p |\ \mu_1\ \dots\ \mu_i\ P\ \mu_{i+1}\ \dots\ \mu_{2N}\ | q \rangle ={}&
\begin{cases}
\langle p |\ \mu_1\ \dots\ \mu_i\ |P]\ \langle P| \mu_{i+1}\ \dots\ \mu_{2N}\ | q \rangle & i \text{ even}\\
\langle p |\ \mu_1\ \dots\ \mu_i\ |P\rangle\ \big(\langle P| \mu_{i+1}\ \dots\ \mu_{2N}\ | q ]\big)^* & i \text{ odd}
\end{cases}\,,\\
\label{eq:p_in_square}
[ p |\ \mu_1\ \dots\ \mu_i\ P\ \mu_{i+1}\ \dots\ \mu_{2N}\ | q ] ={}&
\begin{cases}
\big(\langle p |\ \mu_1\ \dots\ \mu_i\ |P]\big)^* \ [ P| \mu_{i+1}\ \dots\ \mu_{2N}\ | q ] & i \text{ even}\\
[ p |\ \mu_1\ \dots\ \mu_i\ |P]\ \langle P| \mu_{i+1}\ \dots\ \mu_{2N}\ | q ] & i \text{ odd}
\end{cases}\,.
\end{align}
\endgroup
For contractions of vector currents we can use the Fierz identity
\begin{equation}
\label{eq:Fierz}
\langle p|\ \mu\ |q]\ \langle k|\ \mu\ |l] = 2 \spa p.k \spb l.q\,.
\end{equation}
The scalar angle and square products are given by
\begin{align}
\label{eq:angle_product}
\spa p.q ={}& {\big(u^-(p)\big)}^\dagger u^+(q) =
\sqrt{p^-q^+}\hat{p}_{i,\perp} - \sqrt{p^+q^-}\hat{q}_{j,\perp} = - \spa q.p\,,\\
\label{eq:square_product}
\spb p.q ={}& {\big(u^+(p)\big)}^\dagger u^-(q) = -\spa p.q ^* = - \spb q.p\,.
\end{align}
We also define polarisation vectors
\begin{equation}
\label{eq:pol_vector}
\epsilon_\mu^-(p_g, p_r) = \frac{\langle p_r|\mu|p_g]}{\sqrt{2}\spa p_g.{p_r}}\,,\qquad\epsilon_\mu^+(p_g, p_r) = \frac{\langle p_g|\mu|p_r]}{\sqrt{2}\spb p_g.{p_r}}\,.
\end{equation}
fulfilling
\begin{equation}
\label{eq:pol_vector_norm}
\epsilon_\mu^\lambda(p_g, p_r)\big[\epsilon^{\mu\,\lambda'}(p_g, p_r)\big]^* = -\delta_{\lambda\lambda'}\,.
\end{equation}
\subsubsection{Contraction algorithm}
\label{sec:contr_calc_algo}
The contractions are now calculated as follows:
\begin{enumerate}
\item Use equations \eqref{eq:jplus_gen}, \eqref{eq:current_spinhel} to write all currents in a canonical form.
\item Assume that all momenta are lightlike and use the relations
\eqref{eq:p_in_current}, \eqref{eq:p_in_angle}, \eqref{eq:p_in_square}
to split up currents that are contracted with momenta.
\item Apply the Fierz transformation~\eqref{eq:Fierz} to eliminate
contractions between vector currents.
\item Write the arguments of the antisymmetric angle and scalar products in canonical order, see equations~\eqref{eq:angle_product} ,\eqref{eq:square_product}.
\end{enumerate}
The corresponding \lstinline!ContractCurrents! procedure is implemented in
\texttt{include/helspin.fm}.
\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.hepforge.org/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.hepforge.org/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 number of \lstinline!HEJ::Analysis! objects 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!.
\subsubsection{Outgoing parton momentum generation}
\label{sec:out_parton_momenta}
To generate the parton momenta, we use flat sampling in
rapidity and azimuthal angle. The scalar transverse momenta is
generated based on a random variable $x_{p_\perp}$ according to
\begin{equation}
\label{eq:pt_sampling}
p_\perp = p_{\perp,\text{min}} +
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)\,,
\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 $p_{\perp,\text{par}}$
is a generation parameter set to the heuristically determined value of
\begin{equation}
\label{eq:pt_par}
p_{\perp,\text{par}}=p_{\perp,\min}+\frac{n_p}{5}.
\end{equation}
If there are no outgoing bosons, the transverse momentum of the last
parton is fixed by transverse momentum conservation and we only
-generate the rapidity and azimuthal angle.
+generate its rapidity and azimuthal angle.
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. As a more stringent
test, we then check that all partons form separate jets.
\subsubsection{Boson momentum generation}
\label{sec:out_boson}
Next, we generate any potential colourless emissions.\todo{Implement
$WW$, see gitlab \#186} The transverse momentum components are fixed
via momentum conservation. For vector bosons\todo{Higgs} we sample the
rapidity as a Gaussian centred around $|y| = 2.6$ with a standard
deviation of $\Delta y = 1.8$. We randomly decide between forward and
backward direction. The rationale behind this sampling is that the
boson predominantly follows the direction of the emitting parton,
which for LL configurations is an extremal quark or antiquark. To
account for off-shell production, we generate the $p^2$ of the boson
according to a Breit-Wigner distribution.\todo{weight for Higgs}
If decay products are given either in the process definition or
separately in the configuration, we then generate their momenta. If
-several decay channels are possible we choose a random one according
-to the partial widths.
+several decay channels are possible we choose a random one, with the
+different options weighted the partial widths.
Only decays into two particles are implemented. We choose polar and
azimuthal angle of the first decay product by flat sampling in the
parent rest frame. The energy and the momentum of the second decay
product are fixed by momentum conservation. Finally, we boost the
momenta to the lab frame.
\subsubsection{Incoming parton generation}
\label{sec:in_partons}
We then determine the incoming momenta and flavours in
\lstinline!PhaseSpacePoint::reconstruct_incoming!. The four-momenta are given by
\begin{align}
\label{eq:pa}
p_a ={}& (E_a, \mathbf{p_a}) = \frac{\sqrt{s}}{2}(x_a,0,0,-x_a),\\
\label{eq:pb}
p_b ={}& (E_b, \mathbf{p_b}) = \frac{\sqrt{s}}{2}(x_b,0,0,x_b),
\end{align}
where $x_a, x_b$ are calculated as in
eqs.~\eqref{eq:x_a},~\eqref{eq:x_b}.
To generate the incoming flavours, we first determine which ones are
allowed. Initially, all light flavours are allowed unless a specific
flavour is chosen in the process definition in the HEJFOG
configuration file. We then apply the following restrictions
\begin{enumerate}
\item If there is an outgoing vector boson (photon, Z, W$^{\pm}$) we
need a quark or antiquark to couple to.
\begin{itemize}
\item If there are four or more jets
and central $q\bar{q}$ emission is allowed, the boson can couple to
the outgoing $q$ or $\bar{q}$ and there is no additional constraint on the
incoming flavours.
\item Otherwise, there has to be an incoming quark or antiquark. For a
W emission, the sign of the fermion and W boson charges have to match and
\item If there are three or more jets and extremal $q\bar{q}$ emission
is allowed, it is also possible to have an incoming gluon instead of a
quark/antiquark.
\end{itemize}
\item If the settings in the configuration file enforce extremal
$q\bar{q}$ emission we restrict one of the incoming flavours to be a
gluon. Likewise, if the configuration file only allows unordered gluon
emission, we ensure that one of the incoming flavours is a quark or
antiquark.
\end{enumerate}
The constraints are applied as follows. If one of the incoming
flavours automatically fulfills the constraint, then we ignore the
constraint. As an example, if we consider W + jets production and the
first incoming parton is set to an up-quark in the configuration file,
then the vector boson emission does not restrict the flavour choice of
the second incoming parton. Conversely, if one of the incoming
flavours \emph{never} fulfills the constraint, we apply it to the
other incoming flavour. In all other cases we randomly apply the
constraint to only the first, only the second, or both of the incoming
partons.
For each incoming proton, we then choose a random flavour according to
its pdf weight.\todo{factor $4/9$ for fermions} Since the
factorisation scale is not yet known at this point, we somewhat
arbitrarily choose the minimum jet transverse momentum to determine
the pdf weights.
\subsubsection{Outgoing parton flavours}
\label{sec:out_parton_flavours}
Initially, we choose a LL configuration: the most backward/forward
outgoing flavours are set to the backward/forward incoming ones and
all other outgoing partons are gluons. In order to decide whether to
-modify any of these flavours, we first determine the set NLL
+modify any of these flavours, we first determine the set of NLL
configurations that are compatible with the configuration file and the
current flavours. All NLL configurations require three or more jets.
\todo[inline]{Differences between code and description}
\begin{itemize}
\item An unordered configuration is possible if
\begin{itemize}
\item The extremal particle is not a Higgs boson.
\item The extremal parton is a quark or antiquark.
- \item The next parton is a gluon.
+ \item The next parton is a gluon. This condition is automatically fulfilled.
\end{itemize}
\item Extremal $q\bar{q}$ emission is possible if both the most
extremal and the following parton are gluons.
\item Central $q\bar{q}$ emission is possible if there are two
- consecutive gluons between the extremal partons.
+ consecutive gluons between the extremal partons. In principle, this is
+ always the case, but we check explicitly.
\item If there is an outgoing vector boson and none of the current
outgoing partons can couple to it, we have to enforce extremal or
central $q\bar{q}$ emission.
\end{itemize}
If there are no allowed NLL configurations, we leave flavours as they
are. Otherwise, we randomly decide whether to adjust flavours to a NLL
configuration based on the probability in the configuration file. If
we do, we choose one of the allowed channels by flat sampling and adjust the flavours as follows.
\begin{itemize}
\item For unordered emission, we swap the flavours of the most
extremal and the following parton. If both backward and forward
unordered gluons are possible, we choose one at random.
\item For central $q\bar{q}$ emission we choose a random pair of
neigbouring gluons excepting extremal partons. We assign a randomly
chosen quark or antiquark flavour to the first gluon and the
corresponding antiparticle to the second one. We use flat sampling in both cases.
\item For extremal $q\bar{q}$ emission we assign a randomly chosen
quark or antiquark flavour to an extremal gluon and the corresponding
antiparticle to the next gluon. If both backward and forward emission
are possible we again choose randomly.
\end{itemize}
For W processes where the W boson can only couple to a quark or
antiquark from central or extremal $q\bar{q}$ we exclude the strange
quark and antiquark from the flavour sampling.
Finally, again for the case of W boson production, we identify all
quarks and antiquarks the W boson can couple to, and change the
flavour to the other (anti)quark in the same generation with
opposite-sign charge.
\input{currents}
\appendix
\section{Continuous Integration}
\label{sec:CI}
Whenever you are implementing something new or fixed a bug, please also add a
test for the new behaviour to \texttt{t/CMakeLists.txt} via
\lstinline!add_test!. These test can be triggered by running
\lstinline!make test! or \lstinline!ctest! after compiling. A typical test
should be at most a few seconds, so it can be potentially run on each commit
change by each developer. If you require a longer, more careful test, preferably
on top of a small one, surround it with
\begin{lstlisting}[caption={}]
if(${TEST_ALL})
add_test(
NAME t_feature
COMMAND really_long_test
)
endif()
\end{lstlisting}
Afterwards you can execute the longer tests with\footnote{No recompiling is
needed, as long as only the \lstinline!add_test! command is guarded, not the
compiling commands itself.}
\begin{lstlisting}[language=sh,caption={}]
cmake base/directory -DTEST_ALL=TRUE
make test
\end{lstlisting}
On top of that you should add
\href{https://en.cppreference.com/w/cpp/error/assert}{\lstinline!assert!s} in
the code itself. They are only executed when compiled with
\lstinline!CMAKE_BUILD_TYPE=Debug!, without slowing down release code. So you
can use them everywhere to test \textit{expected} or \textit{assumed} behaviour,
e.g. requiring a Higgs boson or relying on rapidity ordering.
GitLab provides ways to directly test code via \textit{Continuous integrations}.
The CI is controlled by \texttt{.gitlab-ci.yml}. For all options for the YAML
file see \href{https://docs.gitlab.com/ee/ci/yaml/}{docs.gitlab.com/ee/ci/yaml/}.https://gitlab.dur.scotgrid.ac.uk/hej/docold/tree/master/Theses
GitLab also provides a small tool to check that YAML syntax is correct under
\lstinline!CI/CD > Pipelines > CI Lint! or
\href{https://gitlab.dur.scotgrid.ac.uk/hej/HEJ/-/ci/lint}{gitlab.dur.scotgrid.ac.uk/hej/HEJ/-/ci/lint}.
Currently the CI is configured to trigger a \textit{Pipeline} on each
\lstinline!git push!. The corresponding \textit{GitLab runners} are configured
under \lstinline!CI/CD Settings>Runners! in the GitLab UI. All runners use a
\href{https://www.docker.com/}{docker} image as virtual environments\footnote{To
use only Docker runners set the \lstinline!docker! tag in
\texttt{.gitlab-ci.yml}.}. The specific docker images maintained separately. If
you add a new dependencies, please also provide a docker image for the CI. The
goal to be able to test \HEJ with all possible configurations.
Each pipeline contains multiple stages (see \lstinline!stages! in
\texttt{.gitlab-ci.yml}) which are executed in order from top to bottom.
Additionally each stage contains multiple jobs. For example the stage
\lstinline!build! contains the jobs \lstinline!build:basic!,
\lstinline!build:qcdloop!, \lstinline!build:rivet!, etc., which compile \HEJ for
different environments and dependencies, by using different Docker images. Jobs
starting with an dot are ignored by the Runner, e.g. \lstinline!.HEJ_build! is
only used as a template, but never executed directly. Only after all jobs of the
previous stage was executed without any error the next stage will start.
To pass information between multiple stages we use \lstinline!artifacts!. The
runner will automatically load all artifacts form all \lstinline!dependencies!
for each job\footnote{If no dependencies are defined \textit{all} artifacts from
all previous jobs are downloaded. Thus please specify an empty dependence if you
do not want to load any artifacts.}. For example the compiled \HEJ code from
\lstinline!build:basic! gets loaded in \lstinline!test:basic! and
\lstinline!FOG:build:basic!, without recompiling \HEJ again. Additionally
artifacts can be downloaded from the GitLab web page, which could be handy for
debugging.
We also trigger some jobs \lstinline!only! on specific events. For example we
only push the code to
\href{https://phab.hepforge.org/source/hej/repository/v2.0/}{HepForge} on
release branches (e.g. v2.0). Also we only execute the \textit{long} tests for
merge requests, on pushes for any release or the \lstinline!master! branch, or
when triggered manually from the GitLab web page.
The actual commands are given in the \lstinline!before_script!,
\lstinline!script! and \lstinline!after_script!
\footnote{\lstinline!after_script! is always executed} sections, and are
standard Linux shell commands (dependent on the docker image). Any failed
command, i.e. returning not zero, stops the job and making the pipeline fail
entirely. Most tests are just running \lstinline!make test! or are based on it.
Thus, to emphasise it again, write tests for your code in \lstinline!cmake!. The
CI is only intended to make automated testing in different environments easier.
\section{Monte Carlo uncertainty}
\label{sec:MC_err}
Since \HEJ is reweighting each Fixed Order point with multiple resummation
events, the Monte Carlo uncertainty of \HEJ is a little bit more complicated
then usual. We start by defining the \HEJ cross section after $N$ FO points
\begin{align}
\sigma_N:=\sum_{i}^N x_i \sum_{j}^{M_i} y_{i,j}=:\sum_i^N\sum_j^{M_i} w_{i,j},
\end{align}
where $x_i$ are the FO weights\footnote{In this definition $x_i$ can be zero,
see the discussion in the next section.}, $y_{i,j}$ are the reweighting weights
, and $M_i$ the number of resummation points. We can set $M=M_i \forall i$ by
potentially adding some points with $y_{i,j}=0$, i.e. $M$ correspond to the
\lstinline!trials! in \lstinline!EventReweighter!. $w_{i,j}$ are the weights as
written out by \HEJ. The expectation value of $\sigma$ is then
\begin{align}
\ev{\sigma_N}= \sum_i \ev{x_i}\sum_j\ev{y_{i,j}}=M \mu_x\sum_i\mu_{y_i},\label{eq:true_sigma}
\end{align}
with $\mu_{x/y}$ being the (true) mean value of $x$ or $y$, i.e.
\begin{align}
\mu_{x}:=\ev{\bar{x}}=\ev{\frac{\sum_i x_i}{N}}=\ev{x}.
\end{align}
The true underlying standard derivation on $\sigma_N$, assuming $\delta_{x}$
and $\delta_{y_i}$ are the standard derivations of $x$ and $y_i$ is
\begin{align}
\delta_{\sigma_N}^2&=M^2 \delta_{x}^2 \sum_i \mu_{y_i}^2
+M \mu_x^2 \sum_i \delta_{y_i}^2. \label{eq:true_err}
\end{align}
Notice that each point $i$ can have an different expectation for $y_i$.
Since we do not know the true distribution of $x$ and $y$ we need to estimate
it. We use the standard derivation
\begin{align}
\tilde{\delta}_{x_i}^2&:=\left(x_i-\bar x\right)^2
=\left(\frac{N-1}{N} x_i - \frac{\sum_{j\neq i} x_j}{N}\right)^2
\label{eq:err_x}\\
\tilde{\delta}_{y_{i,j}}^2&:=\left(y_{i,j}-\bar y_i\right)^2 \label{eq:err_y},
\end{align}
and the mean values $\bar x$ and $\bar y$, to get an estimator for
$\delta_{\sigma_N}$
\begin{align}
\tilde\delta_{\sigma_N}^2&=M^2 \sum_i \tilde\delta_{x_i}^2 \bar{y_i}^2
+\sum_{i,j} x_i^2\tilde\delta_{y_{i,j}}^2. \label{eq:esti_err}
\end{align}
Trough error propagation we can connect the estimated uncertainties back to the
fundamental ones
\begin{align}
\delta_{\tilde{\delta}_{x_i}}^2=\frac{N-1}{N} \delta_x^2.
\end{align}
Together with $\delta_x^2=\ev{x^2}-\ev{x}^2$ and $\ev{\tilde\delta}=0$ this
leads to
\begin{align}
\ev{\tilde{\delta}_{x_i}^2 \bar y_i^2}&=\ev{\tilde{\delta}_{x_i} \bar y_i}^2
+\delta_{\tilde{\delta}_{x_i}}^2 \mu_{y_i}^2
+\delta_{y_i}^2 \mu_{\tilde\delta}^2 \\
&=\frac{N-1}{N} \delta_x^2\mu_{y_i}^2,
\end{align}
and a similar results for $y$. Therefore
\begin{align}
\ev{\delta_{\sigma_N}}=\frac{N-1}{N} M^2 \delta_{x}^2 \sum_i \mu_{y_i}^2
+\frac{M-1}{M} M \mu_x^2 \sum_i \delta_{y_i}^2,
\end{align}
where we can compensate for the additional factors compared to~\eqref{eq:true_err}, by replacing
\begin{align}
\tilde\delta_x&\to\frac{N}{N-1}\tilde\delta_x \label{eq:xcom_bias}\\
\tilde\delta_{y_i}&\to\frac{M}{M-1}\tilde\delta_{y_i}. \label{eq:ycom_bias}
\end{align}
Thus~\eqref{eq:esti_err} is an unbiased estimator of $\delta_{\sigma_N}$.
\subsection{Number of events vs. number of trials}
Even though the above calculation is completely valid, it is unpractical. Both
$x_i$ and $y_{ij}$ could be zero, but zero weight events are typically not
written out. In that sense $N$ and $M$ are the \textit{number of trials} it took
to generate $N'$ and $M'$ (non-zero) events. We can not naively replace all $N$
and $M$ with $N'$ and $M'$ in the above equations, since this would also change
the definition of the average $\bar x$ and $\bar y$.
For illustration let us consider unweighted events, with all weights equal to
$x'$, without changing the cross section $\sum_i^N x_i=\sum_i^{N'} x'_i=N' x'$.
Then the average trial weight is unequal to the average event weight
\begin{align}
\bar x = \frac{\sum_i^{N} x_i}{N} = \frac{\sum_i^{N'} x'}{N}=x'\frac{N'}{N}
\neq x'=\frac{\sum_i^{N'} x'}{N'}.
\end{align}
$N=N'$ would correspond to an $100\%$ efficient unweighting, i.e. a perfect
sampling, where we know the analytical results. In particular using $N'$ instead
of $N$ in the standard derivation gives
\begin{align}
\sum_i \left(x_i-\frac{\sum_i^{N} x_i}{N'}\right)^2=\sum_i \left(x'-x' \frac{\sum_i^{N'}}{N'}\right)^2=0,
\end{align}
which is obviously not true in general for $\tilde\delta^2_x$.
Hence we would have to use the number of trials $N$ everywhere. This would
require an additional parameter to be passed with each events, which is not
always available in practice\footnote{ \texttt{Sherpa} gives the number of
trials, as an \lstinline!attribute::trials! of \lstinline!HEPEUP! in the
\texttt{LHE} file, or similarly as a data member in the HDF5 format
\cite{Hoeche:2019rti}. The \texttt{LHE} standard itself provides the
variable \lstinline!ntries! per event (see
\href{https://phystev.cnrs.fr/wiki/2017:groups:tools:lhe}{this proposal}),
though I have not seen this used anywhere.}. Instead we use
\begin{align}
\tilde\delta_{x}'^2:=\sum_i^{N} x_i^2\geq\tilde\delta_x^2, \label{eq:err_prac}
\end{align}
where the bias of $\delta_x'^2$ vanishes for large $N$. Thus we can use the sum
of weight squares~\eqref{eq:err_prac} instead of~\eqref{eq:err_x}
and~\eqref{eq:err_y}, without worrying about the difference between trials and
generated events. The total error~\eqref{eq:esti_err} becomes
\begin{align}
\tilde\delta_{\sigma_N}^2=\sum_i \left(\sum_j w_{i,j}\right)^2+\sum_{i,j} \left(w_{i,j}\right)^2,
\end{align}
which (conveniently) only dependent on the \HEJ weights $w_{i,j}$.
\section{Explicit formulas for vector currents}
\label{sec:j_vec}
Using eqs.~\eqref{eq:u_plus}\eqref{eq:u_minus}\eqref{eq:jminus_gen}\eqref{eq:Pauli_matrices}\eqref{eq:jplus_gen}, the vector 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)^*\,,\\
j^\pm_\mu(q, p) ={}&\big(j^\pm_\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}
\bibliographystyle{JHEP}
\bibliography{biblio}
\end{document}

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