diff --git a/Changes.md b/Changes.md
index 4cd4d0e..542d109 100644
--- a/Changes.md
+++ b/Changes.md
@@ -1,145 +1,145 @@
# Changelog
This is the log for changes to the HEJ program. Further changes to the HEJ API
are documented in [`Changes-API.md`](Changes-API.md). If you are using HEJ as a
library, please also read the changes there.
## Version 2.1
This release adds support for two new processes:
* W boson with jets.
* Jets with a charged lepton-antilepton pair via a virtual Z boson or photon.
In addition, the complete set of first subleading processes (unordered
gluon, central and extremal quark-antiquark pair) is implemented for
pure jets and W + jets, see
-[arXiv:TODO](https://arxiv.org/abs/TODO). Unordered gluon emission is
-also supported for Higgs boson + jets and Z boson/photon + jets.
+[arXiv:2012.10310](https://arxiv.org/abs/2012.10310). Unordered gluon emission
+is also supported for Higgs boson + jets and Z boson/photon + jets.
This release include many changes to the code, which affect users of
HEJ as a library (see [`Changes-API.md`](Changes-API.md)).
### 2.1.0
#### New Processes
* Resummation for W bosons with jets
- New subleading processes `extremal qqbar` & `central qqbar` for a quark and
anti-quark in the final state, e.g. `g g => u d_bar Wm g` (the other
subleading processes also work with W's)
- `HEJFOG` can generate multiple jets together with a (off-shell) W bosons
decaying into lepton & neutrino
* Resummation for jets with a charged lepton-antilepton pair via a
virtual Z boson or photon. Includes the `unordered` subleading process.
* Resummation can now be performed on all subleading processes within pure
jets also. This includes `unordered`, `extremal qqbar` and `central
qqbar` processes.
#### More Physics implementation
* Partons now have a Colour charge
- Colours are read from and written to LHE files
- For reweighted events the colours are created according to leading colour in
the FKL limit
* Use relative fraction for soft transverse momentum in tagging jets (`soft pt
regulator`) as new (optional) parameter.
- This supersedes `min extparton pt`, which is marked **deprecated** and will
be removed in version 2.2.0
- This is a direct replacement for the old `max ext soft pt fraction`, which
is also **deprecated**.
* Changed the redistribution of the momenta of soft emissions. Now also bosons
can take part of the recoil (previously only jets).
#### Updates to Runcard
* Allow multiplication and division of multiple scale functions e.g.
`H_T/2*m_j1j2`
* Grouped `event treatment` for subleading channels together in runcard
- Rename `non-HEJ` processes to `non-resummable`
* Read electro-weak constants from input
- new mandatory setting `vev` to change vacuum expectation value
- new mandatory settings `particle properties` to specify mass & width of
bosons
- `HEJFOG`: decays are now specified in `decays` setting (previously under
`particle properties`)
* Allow loading multiple analyses with `analyses`. The old `analysis` (with "i")
is marked **deprecated**.
* Optional setting to specify maximal number of Fixed Order events (`max
events`, default is all)
* Allow changing the regulator lambda in input (`regulator parameter`, only for
advanced users)
#### Changes to Input/Output
* Added support to read & write `hdf5` event files suggested in
[arXiv:1905.05120](https://arxiv.org/abs/1905.05120) (needs
[HighFive](https://github.com/BlueBrain/HighFive))
* Support input with average weight equal to the cross section (`IDWTUP=1 or 4`)
* Support unmodified Les Houches Event Files written by Sherpa with
`cross section = sum(weights)/sum(trials)`
* Analyses now get general run information (`LHEF::HEPRUP`) in the
constructor. **This might break previously written, external analyses!**
- external analyses should now be created with
`make_analysis(YAML::Node const & config, LHEF::HEPRUP const & heprup)`
* Support `rivet` version 3 with both `HepMC` version 2 and 3
- Multiple weights with `rivet 3` will only create one `.yoda` file (instead
of one per weight/scale)
* Added option to unweight only resummation events
(`unweight: {type: resummation}`)
* Added option for partially unweighting resummation events, similar to
the fixed-order generator.
* Improved unweighting algorithm.
* Follow HepMC convention for particle Status codes: incoming = 11,
decaying = 2, outgoing = 1 (unchanged)
#### Miscellaneous
* Print cross sections at end of run
* Added example analysis & scale to `examples/`. Everything in `examples/` will
be build when the flag `-DBUILD_EXAMPLES=TRUE` is set in `cmake`.
* Dropped support for HepMC 3.0.0, either HepMC version 2 or >3.1 is required
- It is now possible to write out both HepMC 2 and HepMC 3 events at the same
time
* Require LHADPF version 6. Dropped support for all other versions.
* Use `git-lfs` for raw data in test (`make test` now requires `git-lfs`)
* Currents are now generated with [`FORM`](https://github.com/vermaseren/form)
- `FORM` is included as a `git submodule`, use `git submodule update --init`
to download `FORM`
* Create [Sphinx](http://sphinx-doc.org/) and [Doxygen](http://doxygen.org/)
documentation by `make sphinx` or `make doxygen` in your `build/` folder
## Version 2.0
First release of HEJ 2. Complete code rewrite compared to HEJ 1. Improved
matching to Fixed Order ([arXiv:1805.04446](https://arxiv.org/abs/1805.04446)).
Implemented processes: Higgs boson with jets (FKL and unordered gluon emission,
with finite quark mass loop,
[arXiv:1812.08072](https://arxiv.org/abs/1812.08072)), and pure jets (only FKL).
See [arXiv:1902.08430](https://arxiv.org/abs/1902.08430)
### 2.0.6
* Fixed compiling rivet when YODA headers are _outside_ of rivet directory
### 2.0.5
* Fixed event classification for input not ordered in rapidity
### 2.0.4
* Fixed wrong path of `HEJ_INCLUDE_DIR` in `hej-config.cmake`
### 2.0.3
* Fixed parsing of (numerical factor) * (base scale) in configuration
* Don't change scale names, but sanitise Rivet output file names instead
### 2.0.2
* Changed scale names to `"_over_"` and `"_times_"` for proper file names (was
`"/"` and `"*"` before)
### 2.0.1
* Fixed name of fixed-order generator in error message.
### 2.0.0
* First release
diff --git a/doc/developer_manual/biblio.bib b/doc/developer_manual/biblio.bib
index 82da7ee..71aae00 100644
--- a/doc/developer_manual/biblio.bib
+++ b/doc/developer_manual/biblio.bib
@@ -1,184 +1,194 @@
@Article{Andersen:2008gc,
author = "Andersen, Jeppe R. and Del Duca, Vittorio and White, Chris
D.",
title = "{Higgs Boson Production in Association with Multiple Hard
Jets}",
journal = "JHEP",
volume = "02",
year = "2009",
pages = "015",
eprint = "0808.3696",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
doi = "10.1088/1126-6708/2009/02/015",
SLACcitation = "%%CITATION = 0808.3696;%%"
}
@Article{Andersen:2008ue,
author = "Andersen, Jeppe R. and White, Chris D.",
title = "{A New Framework for Multijet Predictions and its
application to Higgs Boson production at the LHC}",
journal = "Phys. Rev.",
volume = "D78",
year = "2008",
pages = "051501",
eprint = "0802.2858",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
doi = "10.1103/PhysRevD.78.051501",
SLACcitation = "%%CITATION = 0802.2858;%%"
}
@article{Alwall:2006yp,
author = "Alwall, Johan and others",
title = "{A Standard format for Les Houches event files}",
booktitle = "{Monte Carlos for the LHC: A Workshop on the Tools for
LHC Event Simulation (MC4LHC) Geneva, Switzerland, July
17-16, 2006}",
journal = "Comput. Phys. Commun.",
volume = "176",
year = "2007",
pages = "300-304",
doi = "10.1016/j.cpc.2006.11.010",
eprint = "hep-ph/0609017",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FERMILAB-PUB-06-337-T, CERN-LCGAPP-2006-03",
SLACcitation = "%%CITATION = HEP-PH/0609017;%%"
}
@article{Andersen:2017kfc,
author = "Andersen, Jeppe R. and Hapola, Tuomas and Maier, Andreas
and Smillie, Jennifer M.",
title = "{Higgs Boson Plus Dijets: Higher Order Corrections}",
journal = "JHEP",
volume = "09",
year = "2017",
pages = "065",
doi = "10.1007/JHEP09(2017)065",
eprint = "1706.01002",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "EDINBURGH-2017-11, IPPP-17-33, DCPT-17-66, MCNET-17-9",
SLACcitation = "%%CITATION = ARXIV:1706.01002;%%"
}
@article{Andersen:2018tnm,
author = "Andersen, Jeppe R. and Hapola, Tuomas and Heil, Marian
and Maier, Andreas and Smillie, Jennifer M.",
title = "{Higgs-boson plus Dijets: Higher-Order Matching for
High-Energy Predictions}",
year = "2018",
eprint = "1805.04446",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "DCPT/18/66, IPPP/18/33, MCnet-18-10, DCPT-18-66,
IPPP-18-33, MCNET-18-10",
SLACcitation = "%%CITATION = ARXIV:1805.04446;%%"
}
@article{Andersen:2012gk,
author = "Andersen, Jeppe R. and Hapola, Tuomas and Smillie,
Jennifer M.",
title = "{W Plus Multiple Jets at the LHC with High Energy Jets}",
journal = "JHEP",
volume = "09",
year = "2012",
pages = "047",
doi = "10.1007/JHEP09(2012)047",
eprint = "1206.6763",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "EDINBURGH-2012-13, IPPP-12-45, DCPT-12-90,
CP3-ORIGINS-2012-017, DIAS-2012-18, --DIAS-2012-18",
SLACcitation = "%%CITATION = ARXIV:1206.6763;%%"
}
@article{Andersen:2016vkp,
author = "Andersen, Jeppe R. and Medley, Jack J. and Smillie,
Jennifer M.",
title = "{$Z/\gamma^{∗}$ plus multiple hard jets in high energy
collisions}",
journal = "JHEP",
volume = "05",
year = "2016",
pages = "136",
doi = "10.1007/JHEP05(2016)136",
eprint = "1603.05460",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "EDINBURGH-2016-03, IPPP-16-19, DCPT-16-38, MCNET-16-08",
SLACcitation = "%%CITATION = ARXIV:1603.05460;%%"
}
@article{DelDuca:2003ba,
author = "Del Duca, V. and Kilgore, W. and Oleari, C. and Schmidt,
C.R. and Zeppenfeld, D.",
title = "{Kinematical limits on Higgs boson production via gluon
fusion in association with jets}",
journal = "Phys.Rev.",
volume = "D67",
pages = "073003",
doi = "10.1103/PhysRevD.67.073003",
year = "2003",
eprint = "hep-ph/0301013",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "DCPT-02-148, DFTT-20-2002, IPPP-02-74, MADPH-02-1276,
MSUHEP-20620",
SLACcitation = "%%CITATION = HEP-PH/0301013;%%",
}
@article{DelDuca:2001fn,
author = "Del Duca, V. and Kilgore, W. and Oleari, C. and Schmidt,
C. and Zeppenfeld, D.",
title = "{Gluon fusion contributions to H + 2 jet production}",
journal = "Nucl.Phys.",
volume = "B616",
pages = "367-399",
doi = "10.1016/S0550-3213(01)00446-1",
year = "2001",
eprint = "hep-ph/0108030",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "MADPH-01-1235, BNL-HET-01-28, MSUHEP-10709, DFTT-19-2001",
SLACcitation = "%%CITATION = HEP-PH/0108030;%%",
}
@article{Andersen:2011zd,
author = "Andersen, Jeppe R. and Lonnblad, Leif and Smillie,
Jennifer M.",
title = "{A Parton Shower for High Energy Jets}",
journal = "JHEP",
volume = "07",
year = "2011",
pages = "110",
doi = "10.1007/JHEP07(2011)110",
eprint = "1104.1316",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CERN-PH-TH-2011-072, CP3-ORIGINS-2011-14,
EDINBURGH-2011-16, LU-TP-11-15, MCNET-11-12, LU-TP
--11-15",
SLACcitation = "%%CITATION = ARXIV:1104.1316;%%"
}
@article{Andersen:2017sht,
author = "Andersen, Jeppe R. and Brooks, Helen M. and Lönnblad,
Leif",
title = "{Merging High Energy with Soft and Collinear Logarithms
using HEJ and PYTHIA}",
journal = "JHEP",
volume = "09",
year = "2018",
pages = "074",
doi = "10.1007/JHEP09(2018)074",
eprint = "1712.00178",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "CPT/17/184, IPPP/17/92, LU-TP 17-38, MCnet-17-22,
CoEPP-MN-17-22, CPT-17-184, IPPP-17-92, LU-TP-17-38,
MCNET-17-22, COEPP-MN-17-22",
SLACcitation = "%%CITATION = ARXIV:1712.00178;%%"
}
@article{Hoeche:2019rti,
author = "Höche, Stefan and Prestel, Stefan and Schulz, Holger",
title = "{Simulation of vector boson plus many jet final states at
the high luminosity LHC}",
year = "2019",
eprint = "1905.05120",
archivePrefix = "arXiv",
primaryClass = "hep-ph",
reportNumber = "FERMILAB-PUB-19-192-T, LU-TP 19-14, MCNET-19-09",
SLACcitation = "%%CITATION = ARXIV:1905.05120;%%"
}
+@article{Andersen:2020yax,
+ author = "Andersen, Jeppe R. and Black, James A. and Brooks, Helen M. and Byrne, Emmet P. and Maier, Andreas and Smillie, Jennifer M.",
+ title = "{Combined subleading high-energy logarithms and NLO accuracy for W production in association with multiple jets}",
+ eprint = "2012.10310",
+ archivePrefix = "arXiv",
+ primaryClass = "hep-ph",
+ reportNumber = "DESY-20-233, DCPT/20/134, IPPP/20/67, MCnet-20-27, SAGEX-20-28",
+ month = "12",
+ year = "2020"
+}
diff --git a/doc/developer_manual/currents.tex b/doc/developer_manual/currents.tex
index e142aff..6ed450f 100644
--- a/doc/developer_manual/currents.tex
+++ b/doc/developer_manual/currents.tex
@@ -1,717 +1,717 @@
\section{Currents}
\label{sec:currents_impl}
The following section contains a list of all the currents implemented
in \HEJ. Clean up of the code structure is ongoing. All $W$+Jet
currents are located in \texttt{src/Wjets.cc}, all Higgs+Jets currents
are defined in \texttt{src/Hjets.cc}, Z/$\gamma$ + Jet currents are in
\texttt{src/Zjets.cc} and pure jet currents are defined in in
\texttt{src/jets.cc}. All of these have their own separate header
files: \texttt{include/HEJ/Wjets.hh}, \texttt{include/HEJ/Hjets.hh},
\texttt{include/HEJ/Zjets.hh} and \texttt{include/HEJ/jets.hh}
respectively.
The naming convention for the current contraction $\left\|S_{f_1 f_2\to f_1
f_2}\right\|^2$ is \lstinline!ME_[Boson]_[subleading-type]_[incoming]!. For
example \lstinline!ME_W_unob_qq! corresponds to the contraction $j_W^\mu
j_{\text{uno}, \mu}$ ($qQ\to \bar{q}WQg$). For bosons on the same side as the
subleading we drop the connecting underscore, e.g. \lstinline!ME_Wuno_qq!
gives $j_{W,\text{uno}}^\mu j_\mu$ ($qQ\to g\bar{q}WQ$).
\subsection{Pure Jets}
\subsubsection{Quark}
\label{sec:current_quark}
\begin{align}
j_\mu(p_i,p_j)=\bar{u}(p_i)\gamma_\mu u(p_j)
\end{align}
The exact form depends on the helicity and direction (forward/backwards) for the quarks.
\subsubsection{Gluon}
In \HEJ the currents for gluons and quarks are the same, up to a colour factor $K_g/C_F$, where
\begin{align}
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}.
\end{align}
Thus we can just reuse the results from sec.~\ref{sec:current_quark}.
\subsubsection{Single unordered gluon}
Configuration $q(p_a) \to g(p_1) q(p_2) g^*(\tilde{q}_2)$~\cite{Andersen:2017kfc}
\begin{align}
\label{eq:juno}
\begin{split}
&j^{{\rm uno}\; \mu\ cd}(p_2,p_1,p_a) = i \varepsilon_{1\nu} \left( T_{2i}^{c}T_{ia}^d\
\left(U_1^{\mu\nu}-L^{\mu\nu} \right) + T_{2i}^{d}T_{ia}^c\ \left(U_2^{\mu\nu} +
L^{\mu\nu} \right) \right). \\
U_1^{\mu\nu} &= \frac1{s_{21}} \left( j_{21}^\nu j_{1a}^\mu + 2 p_2^\nu
j_{2a}^\mu \right) \qquad \qquad U_2^{\mu\nu} = \frac1{t_{a1}} \left( 2
j_{2a}^\mu p_a^\nu - j_{21}^\mu j_{1a}^\nu \right) \\
L^{\mu\nu} &= \frac1{t_{a2}} \left(-2p_1^\mu j_{2a}^\nu+2p_1.j_{2a}
g^{\mu\nu} + (\tilde{q}_1+\tilde{q}_2)^\nu j_{2a}^\mu + \frac{t_{b2}}{2} j_{2a}^\mu \left(
\frac{p_2^\nu}{p_1.p_2} + \frac{p_b^\nu}{p_1.p_b} \right) \right) ,
\end{split}
\end{align}
$j^{{\rm uno}\; \mu}$ is currently not calculated as a separate current, but always as needed for the ME (i.e. in \lstinline!ME_unob_XX!).
\subsubsection{Extremal \texorpdfstring{$q\bar{q}$}{qqbar}}
In Pure jets we also include the subleading process which arises when
an incoming gluon splits into a $q\bar{q}$ pair. This splitting impact
factor is related to the unordered current by simple means of a
crossing symmetry.
\subsubsection{Central \texorpdfstring{$q\bar{q}$}{qqbar}}
The final subleading process type in the Pure Jets case is Central
$q\bar{q}$. In this process type, we have two currents scattering off
of each other, but this time, via an effective vertex, which connects
together two FKL chains. Each FKL chain t-channel gluon splits into a
$q\bar{q}$ and this results in a quark and anti-quark in between the
most forward and backward jets. One can see an example of such a
process in Figure \ref{fig:qqbarcen_example}.
\begin{figure}[ht]
\centering
\includegraphics[]{Cenqqbar_jx}
\caption{Momentum labeling for a central $q\bar{q}$ process.}
\label{fig:qqbarcen_example}
\end{figure}
As the new central $q\bar{q}$ piece contains the quark propagator, we will treat
this as part of the skeleton process. This means that we do not impose strong ordering
between the $q\bar{q}$-pair taking
\begin{align}
\label{eq:cenqqbarraporder}
y_1 \ll y_q,y_{\bar{q}} \ll y_n.
\end{align}
The HEJ Matrix element for this process can be calculated as:
\begin{align}
\label{eq:Mcentral}
i\mathcal{M} &= g_s^4 T^d_{1a} T^e_{nb}\ \frac{j_{\mu}(p_a,p_1)\ X^{ab\, \mu
\nu}_{{\rm cen}}(p_q,p_{\bar{q}},q_1,q_3)\
j_{\nu}(p_b,p_n)}{t_{a1}t_{bn}}.
\end{align}
where $X^{\mu \nu}_{\rm cen}$ is given by:
\begin{equation}
\label{eq:Xcen}
\begin{split}
X^{\mu \nu}_{\rm cen} ={}&\frac{f^{ced}T^c_{q\bar{q}}}{s_{q\bar{q}}}
\left(\eta^{\mu \nu} X_{sym}^\sigma + V^{\mu \nu \sigma}_{\bar{q}g} \right)
\bar{u}(p_q) \gamma^\sigma u(p_{\bar{q}}) \\ & \qquad + \frac{i
T^d_{qj}T^e_{j\bar{q}}}{(q_1-p_q)^2} X^{\mu\nu}_{\text{qprop}} - \frac{i
T^e_{qj}T^d_{j\bar{q}}}{(q_1-p_{\bar{q}})^2} X^{\mu\nu}_{\text{crossed}}\,,
\end{split}
\end{equation}
with
\begin{align}
\label{eq:Xsym}
X_{sym}^\sigma ={}& q_1^2 \left(
\frac{p_a^\sigma}{s_{aq} + s_{a\bar{q}}} + \frac{p_1^\sigma}{s_{1q} + s_{1\bar{q}}}
\right) - q_3^2 \left(
\frac{p_b^\sigma}{s_{bq} + s_{b\bar{q}}} + \frac{p_n^\sigma}{s_{nq} + s_{n\bar{q}}}
\right)\,,\\
\label{eq:V3g}
V_{3g}^{\mu\nu\sigma} ={}& (q_1 + p_q + p_{\bar{q}})^\nu \eta^{\mu\sigma}
+ (q_3 - p_q - p_{\bar{q}})^\mu \eta^{\nu\sigma}
- (q_1 + q_3)^\sigma \eta^{\mu\nu}\,,\\
\label{eq:Xqprop}
X^{\mu\nu}_{\text{qprop}} ={}& \frac{\langle p_q | \mu (q_1-p_q) \nu | p_{\bar{q}}\rangle}{(q_1-p_q)^2}\,,\\
\label{eq:Xcrossed}
X^{\mu\nu}_{\text{crossed}} ={}& \frac{\langle p_q | \nu (q_1-p_{\bar{q}}) \mu | p_{\bar{q}}\rangle}{(q_1-p_{\bar{q}})^2}\,,
\end{align}
and $q_3 = q_1 - p_q - p_{\bar{q}}$.
\subsection{Higgs}
Different rapidity orderings \todo{give name of functions}
\begin{enumerate}
\item $qQ\to HqQ/qHQ/qQH$ (any rapidity order, full LO ME) $\Rightarrow$ see~\ref{sec:V_H}
\item $qg\to Hqg$ (Higgs outside quark) $\Rightarrow$ see~\ref{sec:V_H}
\item $qg\to qHg$ (central Higgs) $\Rightarrow$ see~\ref{sec:V_H}
\item $qg\to qgH$ (Higgs outside gluon) $\Rightarrow$ see~\ref{sec:jH_mt}
\item $gg\to gHg$ (central Higgs) $\Rightarrow$ see~\ref{sec:V_H}
\item $gg\to ggH$ (Higgs outside gluon) $\Rightarrow$ see~\ref{sec:jH_mt}
\end{enumerate}
\subsubsection{Higgs gluon vertex}
\label{sec:V_H}
The coupling of the Higgs boson to gluons via a virtual quark loop can be written as
\begin{align}
\label{eq:VH}
V^{\mu\nu}_H(q_1, q_2) = \mathgraphics{V_H.pdf} &=
\frac{\alpha_s m^2}{\pi v}\big[
g^{\mu\nu} T_1(q_1, q_2) - q_2^\mu q_1^\nu T_2(q_1, q_2)
\big]\, \\
&\xrightarrow{m \to \infty} \frac{\alpha_s}{3\pi
v} \left(g^{\mu\nu} q_1\cdot q_2 - q_2^\mu q_1^\nu\right).
\end{align}
The outgoing momentum of the Higgs boson is $p_H = q_1 - q_2$.
As a contraction with two currents this by implemented in \lstinline!cHdot! inside \texttt{src/Hjets.cc}.
The form factors $T_1$ and $T_2$ are then given by~\cite{DelDuca:2003ba}
\begin{align}
\label{eq:T_1}
T_1(q_1, q_2) ={}& -C_0(q_1, q_2)\*\left[2\*m^2+\frac{1}{2}\*\left(q_1^2+q_2^2-p_H^2\right)+\frac{2\*q_1^2\*q_2^2\*p_H^2}{\lambda}\right]\notag\\
&-\left[B_0(q_2)-B_0(p_H)\right]\*\frac{q_2^2}{\lambda}\*\left(q_2^2-q_1^2-p_H^2\right)\notag\\
&-\left[B_0(q_1)-B_0(p_H)\right]\*\frac{q_1^2}{\lambda}\*\left(q_1^2-q_2^2-p_H^2\right)-1\,,\displaybreak[0]\\
\label{eq:T_2}
T_2(q_1, q_2) ={}& C_0(q_1,
q_2)\*\left[\frac{4\*m^2}{\lambda}\*\left(p_H^2-q_1^2-q_2^2\right)-1-\frac{4\*q_1^2\*q_2^2}{\lambda}
-
\frac{12\*q_1^2\*q_2^2\*p_H^2}{\lambda^2}\*\left(q_1^2+q_2^2-p_H^2\right)\right]\notag\\
&-\left[B_0(q_2)-B_0(p_H)\right]\*\left[\frac{2\*q_2^2}{\lambda}+\frac{12\*q_1^2\*q_2^2}{\lambda^2}\*\left(q_2^2-q_1^2+p_H^2\right)\right]\notag\\
&-\left[B_0(q_1)-B_0(p_H)\right]\*\left[\frac{2\*q_1^2}{\lambda}+\frac{12\*q_1^2\*q_2^2}{\lambda^2}\*\left(q_1^2-q_2^2+p_H^2\right)\right]\notag\\
&-\frac{2}{\lambda}\*\left(q_1^2+q_2^2-p_H^2\right)\,,
\end{align}
where we have used the scalar bubble and triangle integrals
\begin{align}
\label{eq:B0}
B_0\left(p\right) ={}& \int \frac{d^dl}{i\pi^{\frac{d}{2}}}
\frac{1}{\left(l^2-m^2\right)\left((l+p)^2-m^2\right)}\,,\\
\label{eq:C0}
C_0\left(p,q\right) ={}& \int \frac{d^dl}{i\pi^{\frac{d}{2}}} \frac{1}{\left(l^2-m^2\right)\left((l+p)^2-m^2\right)\left((l+p-q)^2-m^2\right)}\,,
\end{align}
and the K\"{a}ll\'{e}n function
\begin{equation}
\label{eq:lambda}
\lambda = q_1^4 + q_2^4 + p_H^4 - 2\*q_1^2\*q_2^2 - 2\*q_1^2\*p_H^2- 2\*q_2^2\*p_H^2\,.
\end{equation}
The Integrals as such are provided by \QCDloop{} (see wrapper functions \lstinline!B0DD! and \lstinline!C0DD! in \texttt{src/Hjets.cc}).
In the code we are sticking to the convention of~\cite{DelDuca:2003ba}, thus instead of the $T_{1/2}$ we implement (in the functions \lstinline!A1! and \lstinline!A2!)
\begin{align}
\label{eq:A_1}
A_1(q_1, q_2) ={}& \frac{i}{16\pi^2}\*T_2(-q_1, q_2)\,,\\
\label{eq:A_2}
A_2(q_1, q_2) ={}& -\frac{i}{16\pi^2}\*T_1(-q_1, q_2)\,.
\end{align}
\subsubsection{Peripheral Higgs emission - Finite quark mass}
\label{sec:jH_mt}
We describe the emission of a peripheral Higgs boson close to a
scattering gluon with an effective current. In the following we consider
a lightcone decomposition of the gluon momenta, i.e. $p^\pm = E \pm p_z$
and $p_\perp = p_x + i p_y$. The incoming gluon momentum $p_a$ defines
the $-$ direction, so that $p_a^+ = p_{a\perp} = 0$. The outgoing
momenta are $p_1$ for the gluon and $p_H$ for the Higgs boson. We choose
the following polarisation vectors
\begin{equation}
\label{eq:pol_vectors}
\epsilon_\mu^\pm(p_a) = \epsilon_\mu^\pm(p_a, p_1),\qquad \epsilon_\mu^\pm(p_1) = \epsilon_\mu^\pm(p_1, p_a)\,.
\end{equation} The polarisation vectors with two momentum arguments
are defined in equation~(\ref{eq:pol_vector}).
Following~\cite{DelDuca:2001fn}, we introduce effective polarisation
vectors to describe the contraction with the Higgs-boson production
vertex eq.~\eqref{eq:VH}:
\begin{align}
\label{eq:eps_H}
\epsilon_{H,\mu}(p_a) = \frac{T_2(p_a, p_a-p_H)}{(p_a-p_H)^2}\big[p_a\cdot
p_H\epsilon_\mu(p_a) - p_H\cdot\epsilon(p_a) p_{a,\mu}\big]\,,\\
\epsilon_{H,\mu}^*(p_1) = -\frac{T_2(p_1+p_H, p_1)}{(p_1+p_H)^2}\big[p_1\cdot
p_H\epsilon_\mu^*(p_1) - p_H\cdot\epsilon^*(p_1) p_{1,\mu}\big]\,,
\end{align}
with $T_1$ from equation~\eqref{eq:T_1} and $T_2$ from equation~\eqref{eq:T_2}.
Without loss of generality, we consider only the case where the incoming
gluon has positive helicity. The remaining helicity configurations can
be obtained through parity transformation.
Labelling the effective current by the helicities of the gluons we obtain
for the same-helicity case
\begin{equation}
\label{eq:jH_same_helicity}
\begin{split}
j_{H,\mu}^{++}{}&(p_1,p_a,p_H) =
\frac{m^2}{\pi v}\bigg[\\
&-\sqrt{\frac{2p_1^-}{p_a^-}}\frac{p_{1\perp}^*}{|p_{1\perp}|}\frac{t_2}{\spb a.1}\epsilon^{+,*}_{H,\mu}(p_1)
+\sqrt{\frac{2p_a^-}{p_1^-}}\frac{p_{1\perp}^*}{|p_{1\perp}|}\frac{t_2}{\spa 1.a}\epsilon^{+}_{H,\mu}(p_a)\\
&+ [1|H|a\rangle \bigg(
\frac{\sqrt{2}}{\spa 1.a}\epsilon^{+}_{H,\mu}(p_a)
+ \frac{\sqrt{2}}{\spb a.1}\epsilon^{+,*}_{H,\mu}(p_1)-\frac{\spa 1.a T_2(p_a, p_a-p_H)}{\sqrt{2}(p_a-p_H)^2}\epsilon^{+,*}_{\mu}(p_1)\\
&
\qquad
-\frac{\spb a.1 T_2(p_1+p_H,
p_1)}{\sqrt{2}(p_1+p_H)^2}\epsilon^{+}_{\mu}(p_a)-\frac{RH_4}{\sqrt{2}\spb a.1}\epsilon^{+,*}_{\mu}(p_1)+\frac{RH_5}{\sqrt{2}\spa 1.a}\epsilon^{+}_{\mu}(p_a)
\bigg)\\
&
- \frac{[1|H|a\rangle^2}{2 t_1}(p_{a,\mu} RH_{10} - p_{1,\mu} RH_{12})\bigg]
\end{split}
\end{equation}
with $t_1 = (p_a-p_1)^2$, $t_2 = (p_a-p_1-p_H)^2$ and $R = 8 \pi^2$\todo{Code has $R=8\pi^2 i$}. Like other special currents, eq.~\eqref{eq:jH_same_helicity}
is implemented in \texttt{current\_generator/include/currents.frm}.
The currents with a helicity flip is given through
\begin{equation}
\label{eq:jH_helicity_flip}
\begin{split}
j_{H,\mu}^{+-}{}&(p_1,p_a,p_H) =
\frac{m^2}{\pi v}\bigg[\\
&-\sqrt{\frac{2p_1^-}{p_a^-}}\frac{p_{1\perp}^*}{|p_{1\perp}|}\frac{t_2}{\spb
a.1}\epsilon^{-,*}_{H,\mu}(p_1)
+\sqrt{\frac{2p_a^-}{p_1^-}}\frac{p_{1\perp}}{|p_{1\perp}|}\frac{t_2}{\spb a.1}\epsilon^{+}_{H,\mu}(p_a)\\
&+ [1|H|a\rangle \left(
\frac{\sqrt{2}}{\spb a.1} \epsilon^{-,*}_{H,\mu}(p_1)
-\frac{\spa 1.a T_2(p_a, p_a-p_H)}{\sqrt{2}(p_a-p_H)^2}\epsilon^{-,*}_{\mu}(p_1)
- \frac{RH_4}{\sqrt{2}\spb a.1}\epsilon^{-,*}_{\mu}(p_1)\right)
\\
&+ [a|H|1\rangle \left(
\frac{\sqrt{2}}{\spb a.1}\epsilon^{+}_{H,\mu}(p_a)
-\frac{\spa 1.a
T_2(p_1+p_H,p_1)}{\sqrt{2}(p_1+p_H)^2}\epsilon^{+}_{\mu}(p_a)
+\frac{RH_5}{\sqrt{2}\spb a.1}\epsilon^{+}_{\mu}(p_a)
\right)\\
& - \frac{[1|H|a\rangle [a|H|1\rangle}{2 \spb a.1 ^2}(p_{a,\mu} RH_{10} - p_{1,\mu}
RH_{12})\\
&+ \frac{\spa 1.a}{\spb a.1}\bigg(RH_1p_{1,\mu}-RH_2p_{a,\mu}+2
p_1\cdot p_H \frac{T_2(p_1+p_H, p_1)}{(p_1+p_H)^2} p_{a,\mu}
\\
&
\qquad- 2p_a \cdot p_H \frac{T_2(p_a, p_a-p_H)}{(p_a-p_H)^2} p_{1,\mu}+ T_1(p_a-p_1, p_a-p_1-p_H)\frac{(p_1 + p_a)_\mu}{t_1}\\
&\qquad-\frac{(p_1+p_a)\cdot p_H}{t_1} T_2(p_a-p_1, p_a-p_1-p_H)(p_1 - p_a)_\mu
\bigg)
\bigg]\,,
\end{split}
\end{equation}
and implemented again in \texttt{current\_generator/include/currents.frm}.\todo{sign mismatch in line 5 and negative-helicity polarisation vectors}
If we instead choose the gluon momentum in the $+$ direction, so that
$p_a^- = p_{a\perp} = 0$, the corresponding currents are obtained by
replacing $p_1^- \to p_1^+, p_a^- \to p_a^+,
\frac{p_{1\perp}}{|p_{1\perp}|} \to -1$ in the second line of
eq.~\eqref{eq:jH_same_helicity} and eq.~\eqref{eq:jH_helicity_flip}..
The form factors $H_1,H_2,H_4,H_5, H_{10}, H_{12}$ are given in~\cite{DelDuca:2003ba}, and are implemented as \lstinline!H1DD,H2DD! etc. in \texttt{src/Hjets.cc}. They reduce down to fundamental QCD integrals, which are again provided by \QCDloop.
\subsubsection{Peripheral Higgs emission - Infinite top mass}
\label{sec:jH_eff}
To get the result with infinite top mass we could either take the limit $m_t\to \infty$ in~\eqref{eq:jH_helicity_flip} and~\eqref{eq:jH_same_helicity}, or use the \textit{impact factors} as given in~\cite{DelDuca:2003ba}. Both methods are equivalent, and lead to the same result. For the first one would find
\begin{align}
\lim_{m_t\to\infty} m_t^2 H_1 &= i \frac{1}{24 \pi^2}\\
\lim_{m_t\to\infty} m_t^2 H_2 &=-i \frac{1}{24 \pi^2}\\
\lim_{m_t\to\infty} m_t^2 H_4 &= i \frac{1}{24 \pi^2}\\
\lim_{m_t\to\infty} m_t^2 H_5 &=-i \frac{1}{24 \pi^2}\\
\lim_{m_t\to\infty} m_t^2 H_{10} &= 0 \\
\lim_{m_t\to\infty} m_t^2 H_{12} &= 0.
\end{align}
\todo{double check this, see James thesis eq. 4.33}
However only the second method is implemented in the code through \lstinline!C2gHgp!
and \lstinline!C2gHgm! inside \texttt{src/Hjets.cc}, each function
calculates the square of eq. (4.23) and (4.22) from~\cite{DelDuca:2003ba} respectively.
\subsection{Vector Boson + Jets}
\label{sec:currents_W}
\subsubsection{Quark+ Vector Boson}
\begin{figure}
\centering
\begin{minipage}[b]{0.2\textwidth}
\includegraphics[width=\textwidth]{Wbits.pdf}
\end{minipage}
\begin{minipage}[b]{0.1\textwidth}
\centering{=}
\vspace{0.7cm}
\end{minipage}
\begin{minipage}[b]{0.2\textwidth}
\includegraphics[width=\textwidth]{Wbits2.pdf}
\end{minipage}
\begin{minipage}[b]{0.1\textwidth}
\centering{+}
\vspace{0.7cm}
\end{minipage}
\begin{minipage}[b]{0.2\textwidth}
\includegraphics[width=\textwidth]{Wbits3.pdf}
\end{minipage}
\caption{The $j_V$ current is constructed from the two diagrams which
contribute to the emission of a vector boson from a given quark line.}
\label{fig:jV}
\end{figure}
For a $W, Z$, or photon emission we require a fermion. The current is actually a sum of
two separate contributions, see figure~\ref{fig:jV}, one with a vector boson emission
from the initial state, and one with the vector boson emission from the final state.
This can be seen as the following two
-terms, given for the example of a $W$ emission~\cite{Andersen:2012gk}\todo{cite W subleading paper}:
+terms, given for the example of a $W$ emission~\cite{Andersen:2012gk,Andersen:2020yax}
\begin{align}
\label{eq:Weffcur1}
j_W^\mu(p_a,p_{\ell},p_{\bar{\ell}}, p_1) =&\ \frac{g_W^2}{2}\
\frac1{p_W^2-M_W^2+i\ \Gamma_W M_W}\ \bar{u}^-(p_\ell) \gamma_\alpha
v^-(p_{\bar\ell})\nonumber \\
& \cdot \left( \frac{ \bar{u}^-(p_1) \gamma^\alpha (\slashed{p}_W +
\slashed{p}_1)\gamma^\mu u^-(p_a)}{(p_W+p_1)^2} +
\frac{ \bar{u}^-(p_1)\gamma^\mu (\slashed{p}_a - \slashed{p}_W)\gamma^\alpha u^-(p_a)}{(p_a-p_W)^2} \right).
\end{align}
There are a couple of subtleties here. There is a minus sign
distinction between the quark-anti-quark cases due to the fermion flow
of the propagator in the current. Note that the type of $W$ emission
(+ or -) will depend on the quark flavour, and that the handedness of
the quark-line is given by whether its a quark or anti-quark.
The coupling and propagator factor in eq.~(\ref{eq:Weffcur1}) have to
be adapted depending on the emitted boson. The remaining product of
currents
\begin{equation}
\label{eq:J_V}
J_{\text{V}}^\mu(p_2,p_l,p_{\bar{l}},p_3)=\left( \frac{ \bar{u}_2 \gamma^\nu (\slashed{p}_2 +
\slashed{p}_l +
\slashed{p}_{\bar{l}}) \gamma^\mu u_3}{s_{2l\bar{l}}} - \frac{\bar u_2
\gamma^\mu(\slashed{p}_3 + \slashed{p}_l + \slashed{p}_{\bar{l}}) \gamma^\nu
u_3}{s_{3l\bar{l}}} \right) [\bar{u}_l \gamma_\nu u_{\bar{l}}]
\end{equation}
with $s_{il\bar{l}} = (p_i + p_l +p_{\bar{l}})^2$ is universal. The
implementation is in \texttt{include/currents.frm} inside the
\texttt{current\_generator} (see section~\ref{sec:cur_gen}). To use it
inside \FORM use the place-holder
\lstinline!JV(h1, hl, mu, pa, p1, plbar, pl)!, where \lstinline!h1! is
the helicity of the quark line and \lstinline!hl! the helicity of the
lepton line.
\subsubsection{Vector boson with unordered emission}
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{vuno1}
\caption{}
\label{fig:U1diags}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{vuno2}
\caption{}
\label{fig:U2diags}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{vuno3}
\caption{}
\label{fig:Cdiags}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{vuno4}
\caption{}
\label{fig:Ddiags}
\end{subfigure}
\vspace{0.4cm}
\caption{Examples of each of the four categories of Feynman diagram which
contribute to at leading-order; there are twelve in total. (a) is an example where the gluon and vector
boson are emitted from the same quark line and the gluon comes after the
$t$-channel propagator. In (b), the gluon and vector boson are emitted from
the same quark line and the gluon comes before the $t$-channel proagator.
In (c) the gluon is emitted from the $t$-channel gluon and in (d) the gluon
is emitted from the $b$--$3$ quark line.}
\label{fig:Vunodiags}
\end{figure}
It is necessary to include subleading processes in vector boson + jets
also. Similarly to the pure jet case, the unordered currents are not
calculated separately, and only in the ME functions when required in
the \texttt{src/Wjets.cc} or \texttt{src/Zjets.cc} file. The following shows the derivation of the calculation of
this ME within HEJ. We start with a contraction of two currents:
\begin{equation}
\label{eq:SabsVuno}
S_{qQ\to Vgq^\prime Q} =
j_{V{\rm uno}\,\mu}^d(p_a,p_1,p_2,p_\ell,p_{\bar\ell})\ g^{\mu
\nu}\ T^d_{3b}\ j^{h_b,h_3}_{\nu}(p_b,p_{3}),
\end{equation}
where $j_{V,{\rm uno}}$ is our new unordered current which is is
only non-zero for $h_a=h_1=-$ and hence we have suppressed its
helicity indices. It is derived from the 12 leading-order Feynman
diagrams in the QMRK limit (see figure~\ref{fig:Vunodiags}). Using
$T^m_{ij}$ represent fundamental colour matrices between quark state
$i$ and $j$ with adjoint index $m$ we find
\begin{align}\label{eq:wunocurrent}
\begin{split}
j^{d\,\mu}_{\rm V,uno}(p_a,p_1,p_2,p_\ell,p_{\bar{\ell}}) =& \ i \varepsilon_{\nu}(p_1)\
\bar{u}^-(p_\ell) \gamma_\rho v^-(p_{\bar \ell}) \\ & \quad \times\
\left(T^1_{2i} T^d_{ia} (\tilde U_1^{\nu\mu\rho}-\tilde L^{\nu\mu\rho}) +
T^d_{2i} T^1_{ia} (\tilde U_2^{\nu\mu\rho}+\tilde L^{\nu\mu\rho}) \right),
\end{split}
\end{align}
where expressions for $\tilde U_{1,2}^{\nu\mu\rho}$ and $\tilde L^{\nu\mu\rho}$
are given as:
\begin{align}
\label{eq:U1tensor}
\begin{split}
\tilde U_1^{\nu\mu\rho} ={}&\frac{\langle 2|\nu (\slashed{p}_2+ \slashed{p}_1)\mu (\slashed{p}_a - \slashed{p}_V)\rho P_L |a\rangle }{s_{12}t_{aV}} + \frac{\langle 2|\nu (\slashed{p}_2+ \slashed{p}_1)\rho P_L (\slashed{p}_2+\slashed{p}_1 + \slashed{p}_V)\mu |a\rangle }{s_{12}s_{12V}} \\
&+ \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_V) \nu
(\slashed{p}_1 + \slashed{p}_2+\slashed{p}_V)\mu |a\rangle}{s_{2V}s_{12V}}\,,
\end{split}\\
\label{eq:U2tensor}
\begin{split}
\tilde U_2^{\nu\mu\rho} ={}&\frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_V-\slashed{p}_1)\nu (\slashed{p}_a - \slashed{p}_V)\rho P_L |a\rangle }{t_{aV1}t_{aV}} + \frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_V- \slashed{p}_1)\rho P_L (\slashed{p}_a-\slashed{p}_1) \nu |a\rangle }{t_{a1V}t_{a1}} \\
&+ \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_V) \mu
(\slashed{p}_a-\slashed{p}_1)\nu |a\rangle}{s_{2V}t_{a1}}\,,
\end{split}\\
\label{eq:Ltensor}
\begin{split}
\tilde L^{\nu\mu\rho} ={}& \frac{1}{t_{aV2}}\left[
\frac{\langle 2 | \sigma (\slashed{p}_a-\slashed{p}_V)\rho|a\rangle}{t_{aV}}
+\frac{\langle 2 | \rho (\slashed{p}_2+\slashed{p}_V)\sigma|a\rangle}{s_{2V}}
\right]\\
&\times \left\{\left(\frac{p_b^\nu}{s_{1b}} + \frac{p_3^\nu}{s_{13}}\right)(q_1-p_1)^2g^{\mu\sigma}+(2q_1-p_1)^\nu g^{\mu\sigma} - 2p_1^\mu g^{\nu\sigma} + (2p_1-q_1)^\sigma g^{\mu\nu} \right\}\,,
\end{split}
\end{align}
where $s_{ij\dots} = (p_i + p_j + \dots)^2, t_{ij\dots} = (p_i - p_j -
\dots)^2$ and $q_1 = p_a-p_2-p_V$. This is actually calculated and
used in the code in a much cleaner way as follows:
\begin{align}\label{eq:spinorstringVuno}
S_{qQ\to Vgq^\prime Q} = i\varepsilon_\nu (p_g) \bar{u}^-(p_2)&\gamma_\rho\nu(p_{\bar{q}})\times T^d_{3b} \bar{u}^{h_3}(p_3)\gamma_\mu u^{h_b}(p_b) \times \nonumber \\
&\left( T^1_{2i}T^d_{ia} \left( \tilde{U}_1^{\nu\mu\rho}-\tilde{L}^{\nu\mu\rho}\right)+T^d_{2i}T^1_{ia}\left(\tilde{U}_2^{\nu\mu\rho}+\tilde{L}^{\nu\mu\rho}\right) \right)
\end{align}
If we define the objects:
\begin{align}\label{eq:VunoX}
X &= \varepsilon_\nu(p_g) \left[ \bar{u}^-(p_2)\gamma_\rho\nu(p_{\bar{q}})\right] \left[ \bar{u}^{h_3}(p_3)\gamma_\mu u^{h_b}(p_b)\right] \left( \tilde{U}_1^{\nu\mu\rho}-\tilde{L}^{\nu\mu\rho}\right)\\
Y &= \varepsilon_\nu(p_g) \left[ \bar{u}^-(p_2)\gamma_\rho\nu(p_{\bar{q}})\right] \left[ \bar{u}^{h_3}(p_3)\gamma_\mu u^{h_b}(p_b)\right] \left( \tilde{U}_2^{\nu\mu\rho}+\tilde{L}^{\nu\mu\rho}\right)
\label{eq:WunoY}
\end{align}
then we can rewrite Equation \eqref{eq:spinorstringVuno} in the much simpler form:
\begin{equation}
S_{qQ\to Vgq^\prime Q} = iT^d_{3b} \left( T^{1}_{2i}T^d_{ia} X + T^d_{2i}T^1_{ia} Y \right)
\end{equation}
then, by using:
\begin{align}
\sum_{\text{all indices}}& T^d_{3b}T^e_{b3}T^1_{2i}T^d_{ia}T^e_{ai}T^1_{i2} = \frac{1}{2}C_F^2C_A \\
\sum_{\text{all indices}}& T^d_{3b}T^e_{b3}T^1_{2i}T^d_{ia}T^1_{ai}T^e_{i2} = \frac{1}{2}C_F^2C_A - \frac{1}{4}C_A^2C_F = -\frac{1}{4}C_F
\end{align}
giving then, the spin summed and helicity averaged spinor string as:
\begin{equation}\label{eq:VunoSumAveS}
||\;\bar{S}_{qQ\to Vgq^\prime Q}\;|| = \frac{1}{4N_C^2} \left( \frac{1}{2}C_F^2C_A\left(|X|^2+|Y|^2\right)-\frac{1}{4}C_F\times2\mathrm{Re}\left(XY^*\right)\right)
\end{equation}
\subsubsection{\texorpdfstring{$W$}{W}+Extremal \texorpdfstring{$\mathbf{q\bar{q}}$}{qqbar}}
\todo{Update when included in $Z$ + jets}
The $W$+Jet sub-leading processes containing an extremal $q\bar{q}$ are
related by crossing symmetry to the $W$+Jet unordered processes. This
means that one can simply perform a crossing symmetry argument on
eq.~\ref{eq:wunocurrent} to arrive at the extremal $q\bar{q}$ current
required.We show the basic structure of the extremal $q\bar{q}$
current in figure~\ref{fig:qgimp}, neglecting the $W$-emission for
simplicity.
\begin{figure}
\centering
\includegraphics[width=0.3\textwidth]{{qqbarex_schem}}
\caption{Schematic structure of the $gq \to \bar{Q}Qq$ amplitude in the limit
$y_1 \sim y_2 \ll y_3$}
\label{fig:qgimp}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarex1}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarex2}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarex4}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarex5}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarex3}
\end{subfigure}
\caption{The five tree-level graphs which contribute to the process $gq \to \bar{Q}Qq$.}
\label{fig:qg_qQQ_graphs}
\end{figure}
We can obtain the current for $g\rightarrow W q \bar{q}$ by evaluating
the current for $W$ plus unordered emissions with the normal arguments
$p_a \leftrightarrow -p_1 $ interchanged. This is a non-trivial
statement: due to the minimality of the approximations made, the
crossing symmetry normally present in the full amplitude may be
extended to the factorised current.
We must again note that swapping $p_a \leftrightarrow -p_1$ will lead
to $u$-spinors with momenta with negative energy. These are identical
to $v$-spinors with momenta with positive energy, up to an overall
phase which is common to all terms, and can therefore be neglected.
Mathematically, this is given by:
\begin{align}\label{eq:crossedJ}
j^\mu_{\rm W,g\to Q\bar{Q}}(p_a,p_1,p_2,p_\ell,p_{\bar{\ell}}) =i \varepsilon_{g\nu}
\langle \ell | \rho | \bar \ell \rangle_L
\left(T^1_{2i} T^d_{ia} (\tilde U_{1,X}^{\nu\mu\rho}-\tilde L^{\nu\mu\rho}_X) + T^d_{2i} T^1_{ia} (\tilde U_{2,X}^{\nu\mu\rho}+\tilde L_X^{\nu\mu\rho}) \right),
\end{align}
where the components are now given by
\begin{align}
\label{eq:U1tensorX}
\begin{split}
\tilde U_{1,X}^{\nu\mu\rho} =&\frac{\langle 2|\nu (\slashed{p}_a- \slashed{p}_2)\mu (\slashed{p}_1 + \slashed{p}_W)\rho P_L |1\rangle }{t_{a2}s_{1W}} + \frac{\langle 2|\nu (\slashed{p}_a- \slashed{p}_2)\rho P_L (\slashed{p}_a-\slashed{p}_2 - \slashed{p}_W)\mu |1\rangle }{t_{a2}t_{a2W}} \\
&- \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_W) \nu
(\slashed{p}_a - \slashed{p}_2-\slashed{p}_W)\mu
|1\rangle}{s_{2W}t_{a2W}}\,,
\end{split}\\
\label{eq:U2tensorX}
\begin{split}
\tilde U_{2,X}^{\nu\mu\rho} =&-\frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_W-\slashed{p}_1)\nu (\slashed{p}_1 + \slashed{p}_W)\rho P_L |1\rangle }{t_{aW1}s_{1W}} + \frac{\langle 2|\mu (\slashed{p}_a-\slashed{p}_W- \slashed{p}_1)\rho P_L (\slashed{p}_a-\slashed{p}_1) \nu |1\rangle }{t_{a1W}t_{a1}} \\
&+ \frac{\langle 2|\rho P_L (\slashed{p}_2+ \slashed{p}_W) \mu
(\slashed{p}_a-\slashed{p}_1)\nu |1\rangle}{s_{2W}t_{a1}}\,,
\end{split}\\
\label{eq:LtensorX}
\begin{split}
\tilde L^{\nu\mu\rho}_X &= \frac{1}{s_{W12}}\left[-\frac{\langle 2 |\sigma (\slashed{p}_1 + \slashed{p}_W) \rho P_L | 1\rangle}{s_{1W}} + \frac{\langle 2 |\rho P_L (\slashed{p}_2 + \slashed{p}_W) \sigma | 1\rangle }{s_{2W}} \right] \\
&\vphantom{+\frac{1}{t_{aW2}}}\quad\cdot \left( -\left(
\frac{p_b^\nu}{s_{ab}} + \frac{p_n^\nu}{s_{an}} \right) (q_1+p_a)^2 g^{\sigma\mu}+ g^{\sigma \mu} (2q_1 +p_a)^\nu - g^{\mu \nu}(2p_a+q_1)^\sigma+ 2g^{\nu \sigma}p_a^\mu \right)\,,
\end{split}
\end{align}
where $q_1=-(p_1+p_2+p_W)$. Notice in particular the similarity to the $W$+uno scenario (from which
this has been derived).
\subsubsection{Central \texorpdfstring{$\mathbf{q\bar{q}}$}{qqbar} Vertex}
The final subleading process in the $W$+Jet case is the Central
$q\bar{q}$ vertex. This subleading process does not require an altered
current, but an effective vertex which is contracted with two regular
\HEJ currents. This complexity is dealt with nicely by the \FORM inside the
\texttt{current\_generator/j\_Wqqbar\_j.frm}, which is detailed in
section~\ref{sec:contr_calc}.
The $W$-emission can be from the central effective vertex (scenario
dealt with by the function \lstinline!ME_WCenqqbar_qq! in the file
\texttt{src/Wjets.cc}); or from either of the external quark legs
(scenario dealt with by \lstinline!ME_W_Cenqqbar_qq! in same file). In
the pure jets case, there are 7 separate diagrams which contribute to
this, which can be seen in figure~\ref{fig:qq_qQQq_graphs}. In the $W$+Jets
case, there are then 45 separate contributions.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarcen1}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarcen2}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarcen3}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarcen4}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarcen5}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarcen6}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{qqbarcen7}
\end{subfigure}
\caption{All Feynman diagrams which contribute to $qq' \to qQ\bar{Q}q'$ at
leading order.}
\label{fig:qq_qQQq_graphs}
\end{figure}
The end result is of the effective vertex, after derivation, is:
\begin{align}
\label{eq:EffectiveVertexqqbar}
\begin{split}
V^{\mu\nu}_{\text{Eff}}=&
- i \frac{C_1}{s_{23AB}}\left(X^{\mu\nu\sigma}_{1a}\hat{t_1} + X^{\mu\nu\sigma}_{4b}\hat{t_3} +V^{\mu\nu\sigma}_{3g}\right)J_{\text{V} \sigma}(p_2,p_A,p_B,p_3)
\\
&\quad +iC_2X^{\mu\nu}_{Unc}+iC_3X^{\mu\nu}_{Cro},
\end{split}
\end{align}
where:
\begin{align}
\begin{split}
C_1=&T^e_{1q}T^g_{qa}T^e_{23}T^g_{4b} -
T^g_{1q}T^e_{qa}T^e_{23}T^g_{4b} = f^{egc}T^c_{1a}T^e_{23}T^g_{4b},
\\
C_2=&T^g_{1a}T^g_{2q}T^{g'}_{q3}T^{g'}_{4b}
\\
C_3=&T^g_{1a}T^{g'}_{2q}T^g_{q3}T^{g'}_{4b}
\end{split}
\end{align}
are the colour factors of different contributions and $J_\text{V}$ is
given in equation~(\ref{eq:J_V}).
The following tensor structures correspond to groupings of diagrams in
figure~\ref{fig:qq_qQQq_graphs}.
\begin{eqnarray}
\label{eq:X_1a}
X_{1a}^{\mu\nu\sigma} &=
\frac{-g^{\mu\nu}}{s_{23AB}\hat{t_3}}\left(\frac{p^\sigma_a}{s_{a2} + s_{a3} + s_{aA} + s_{aB}} +
\frac{p^\sigma_1}{s_{12} + s_{13} + s_{1A} + s_{1B}}\right)
\\
\label{eq:X_4b}
X_{4b}^{\mu\nu\sigma}
&=\frac{g^{\mu\nu}}{s_{23AB}\hat{t_1}}\left(\frac{p^\sigma_b}{s_{b2} + s_{b3} + s_{bA} + s_{bB}}+
\frac{p^\sigma_4}{s_{42} + s_{43} + s_{4A} + s_{4B}}\right)
\end{eqnarray}
correspond to the first and second row of diagrams in figure~\ref{fig:qq_qQQq_graphs}.
\begin{align}
\label{eq:3GluonWEmit}
\begin{split}
V^{\mu\nu\sigma}_{3g}=\frac{1}{
\hat{t}_1s_{23AB}\,\hat{t}_3}
\bigg[&\left(q_1+p_2+p_3+p_A+p_B\right)^\nu
g^{\mu\sigma}+
\\
&\quad\left(q_3-p_2-p_3-p_A-p_B\right)^\mu g^{\sigma\nu}-
\\
& \qquad\left(q_1+q_3\right)^\sigma g^{\mu\nu}\bigg]J_{\text{V} \sigma}(p_2,p_A,p_B,p_3)
\end{split}
\end{align}
corresponds to the left diagram on the third row in
figure~\ref{fig:qq_qQQq_graphs}. One notes that all of these
contributions have the same colour factor, and as such we can group
them together nicely before summing over helicities etc. As such, the function
\lstinline!M_sym_W! returns a contraction of the above tensor containing the
information from these 5 groupings of contributions (30 diagrams in total). It
is available through the generated header \texttt{j\_Wqqbar\_j.hh} (see
section~\ref{sec:cur_gen}).
\begin{align}
\label{eq:X_Unc}
\begin{split}
X^{\mu\nu}_{Unc}=\frac{\langle A|\sigma P_L|B\rangle}{\hat{t_1}\hat{t_3}} \bar{u}_2&\left[-\frac{
\gamma^\sigma P_L(\slashed{p}_2+\slashed{p}_A+\slashed{p}_B)\gamma^\mu
(\slashed{q}_3+ \slashed{p}_3)\gamma^\nu}{(s_{2AB})(t_{unc_{2}})}\right.+
\\
&\qquad\left. \frac{\gamma^\mu(\slashed{q}_1-\slashed{p}_2)\gamma^\sigma
P_L(\slashed{q}_3+\slashed{p}_3)\gamma^\nu}{(t_{unc_{1}})(t_{unc_{2}})}\right. +
\\
&\qquad\qquad\left. \frac{\gamma^\mu(\slashed{q}_1-\slashed{p}_2)\gamma^\nu(\slashed{p}_3+\slashed{p}_A+\slashed{p}_B)\gamma^\sigma P_L
}{(t_{unc_1})(s_{3AB})}\right]v_3
\end{split}
\end{align}
corresponds to the diagram on the right of row three in
figure~\ref{fig:qq_qQQq_graphs}. This contribution to the current
contraction can be obtained in the code with the function
\lstinline!M_uncross_W!.
\begin{align}
\begin{split}
\label{eq:X_Cro}
X^{\mu\nu}_{Cro}=\frac{\langle
A|\sigma P_L|B\rangle}{\hat{t_1}\hat{t_3}} \bar{u}_2&\left[-\frac{
\gamma^\nu(\slashed{q}_3+\slashed{p}_2)\gamma^\mu
(\slashed{p}_3+\slashed{p}_A+\slashed{p}_B)\gamma^\sigma P_L}{(t_{cro_1})(s_{3AB})}\right.+
\\
&\qquad\left. \frac{\gamma^\nu(\slashed{q}_3+\slashed{p}_2)\gamma^\sigma
P_L(\slashed{q}_1-\slashed{p}_3)\gamma^\mu}{(t_{cro_{1}})(t_{cro_{2}})}\right.+
\\ &\qquad\qquad\left
. \frac{\gamma^\sigma P_L(\slashed{p}_2+\slashed{p}_A+\slashed{p}_B)\gamma^\nu(\slashed{q}_1-\slashed{p}_3)\gamma^\mu
}{(s_{2AB})(t_{cro_2})}\right]v_3
\end{split}
\end{align}
corresponds to the last diagram in
figure~\ref{fig:qq_qQQq_graphs}. This contribution to the current
contraction can be obtained in the code with the function
\lstinline!M_cross_W!.
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "developer_manual"
%%% End:
diff --git a/include/HEJ/Analysis.hh b/include/HEJ/Analysis.hh
index 9d64c32..b165b38 100644
--- a/include/HEJ/Analysis.hh
+++ b/include/HEJ/Analysis.hh
@@ -1,50 +1,50 @@
/** \file
* \brief Header file for the Analysis interface
*
- * This header contains declarations that faciliate creating custom analyses
+ * This header contains declarations that facilitate creating custom analyses
* to be used with HEJ 2.
* \todo link to user documentation
*
* \authors The HEJ collaboration (see AUTHORS for details)
* \date 2019
* \copyright GPLv2 or later
*/
#pragma once
namespace LHEF {
class HEPRUP;
}
//! Main HEJ 2 Namespace
namespace HEJ {
class Event;
//! Analysis base class
/**
* This is the interface that all analyses should implement,
* i.e. all custom analyses have to be derived from this struct.
*/
struct Analysis {
/**
* @param res_event The event in resummation phase space
* @param FO_event The original fixed-order event
*/
virtual void fill(Event const & res_event, Event const & FO_event) = 0;
//! Decide whether an event passes the cuts
/**
* @param res_event The event in resummation phase space
* @param FO_event The original fixed-order event
* @returns Whether the event passes all cuts
*/
virtual bool pass_cuts(Event const & res_event, Event const & FO_event) = 0;
//! Finalise analysis
/**
* This function is called after all events have been processed and
* can be used for example to print out or save the results.
*/
virtual void finalise() = 0;
virtual ~Analysis() = default;
};
} // namespace HEJ