# Using built-in matrix elements¶

There are a wide range of built-in matrix elements included in Herwig. There are examples of using all the matrix elements commented out in the input files for the relevant collider.

Normally an instance of each matrix element class has been created and a standard process selected for each collider, with examples of using the other matrix elements provided in the input file. For example for the LHC from LHC.in

#
#  Electroweak boson W/Z processes
#
# Drell-Yan Z/gamma
insert SubProcess:MatrixElements[0] MEqq2gZ2ff
#
# Drell-Yan W
# insert SubProcess:MatrixElements[0] MEqq2W2ff
#
# W+jet
# insert SubProcess:MatrixElements[0] MEWJet
#
# Z+jet
# insert SubProcess:MatrixElements[0] MEZJet
#
# WW/WZ/ZZ
# insert SubProcess:MatrixElements[0] MEPP2VV
#
# Wgamma/Zgamma
# insert SubProcess:MatrixElements[0] MEPP2VGamma
#

The default process is $$q\bar{q}\to \gamma/Z^0\to f\bar{f}$$ with a range of other processes available by simply commenting in/out the relevant lines (these are only the electroweak processes from LHC.in). There are usually a range of switches which control exactly which process is generated. For example for the default MEqq2gZ2ff there are four switches which control the process generation:
• Process controls which outgoing particles, the decay products of the $$Z^0$$, are generated. By default the ChargedLeptons option which only includes $$e^+e^-$$, $$\mu^+\mu^-$$, and $$\tau^+\tau^-$$ is used;
• MaxFlavour controls the maximum flavour of the incoming quark-antiquark pair, 1-6 for down, up, strange, charm, bottom and top as in the PDG numbering scheme, by default it is set to 5 to ensure all quarks other than top are used;
• MinFlavour controls the minimum flavour of the incoming quark-antiquark pair, by default it is set to 1;
• GammaZ controls whether the photon, $$Z^0$$, or both are included as s-channel intermediates, by default it is set to All so both are included.

The full range of matrix elements included is described briefly below for different types of collisions. The full documentation includes the various switches available for the different classes.

• Matrix Elements for hadron-hadron collisions. There are two example input files provided for hadron-hadron collisions TVT.in for the Tevatron and LHC.in for the LHC.
• Processes involving electroweak, $$W^\pm$$ and $$Z^0$$ bosons
• The MEqq2gZ2ff class implements the process $$q\bar{q}\to \gamma/Z^0\to f\bar{f}$$ with a default instance called MEqq2gZ2ff. This includes the matrix-element correction for hard QCD radiation in the parton shower.
• The MEqq2W2ff class implements the process $$q\bar{q'}\to W^\pm\to f\bar{f'}$$ with a default instance called MEqq2W2ff. This includes the matrix-element correction for hard QCD radiation in the parton shower.
• The MEPP2WJet class implements the process the matrix elements for the production of $$W^\pm+\mathrm{jets}$$ in hadron-hadron collisions, i.e. the processes $$q\bar{q'}\to W^\pm g$$, $$qg\to W^\pm q'$$, $$\bar{q}g\to W^\pm\bar{q'}$$. The default instance is called MEWJet.
• The MEPP2ZJet class implements the process the matrix elements for the production of $$Z^0+\mathrm{jets}$$ in hadron-hadron collisions, i.e. the processes $$q\bar{q}\to Z^0g$$, $$qg\to Z^0q$$, $$\bar{q}g\to Z^0\bar{q}$$. The default instance is called MEZJet.
• The MEPP2VV class implements the production of pairs of massive electroweak gauge bosons, i.e. the processes $$q\bar{q}\to Z^0Z^0$$, $$q\bar{q}\to W^+W^-$$, $$q\bar{q'}\to W^\pm Z^0$$. The default instance is called MEPP2VV.
• The MEPP2VGamma class implements the production of a massive electroweak gauge boson in association with a photon, i.e. the processes $$q\bar{q}\to Z^0\gamma$$ and $$q\bar{q'}\to W^\pm\gamma$$. The default instance is called MEPP2VGamma.
• QCD and electromagnetic processes
• The MEQCD2to2 class implements QCD $$2\to2$$ scattering processes, i.e. the partonic processes $$gg\to gg$$, $$gg\to q\bar{q}$$, $$q\bar{q}\to gg$$, $$gq\to gq$$, $$g\bar{q}\to g\bar{q}$$, $$qq\to qq$$, $$q\bar{q}\bar{q}\to \bar{q}\bar{q}$$ and $$q \bar{q}\to q \bar{q}$$. The default instance is called MEQCD2to2. The masses of the quarks are neglected in the calculation of the matrix elements
• The MEPP2GammaJet class implements the production of a photon in association with a jet, i.e. the partonic processes, $$qg\to q\gamma$$, $$\bar{q}g\to\bar{q}\gamma$$ and $$q\bar{q}\to\gamma g$$. The default instance is called MEGammaJet.
• The MEPP2GammaGamma class implements the production of photon pairs, i.e. the partonic processes $$q\bar{q}$$ and $$gg \to \gamma\gamma$$ where the gluon-gluon process is mediated by a quark loop. The default instance is called MEGammaGamma.
• Processes involving heavy quarks
• The MEPP2QQ class implements the production of heavy quarks including the quark mass effects, i.e. $$q\bar{q}, gg \to Q\bar{Q}$$. The default instance is called MEHeavyQuark.
• The MEPP2SingleTop class implements the production of a single top quark via the $$bq \to tq'$$ t-channel process, the $$q\bar{q'}\to W^\pm \to t\bar{b}, \bar{t}b$$, s-channel $$W^\pm$$ process, and the $$bg\to t W^-, \bar{b}g\to \bar{t}W^+$$ process. There is a default instance created handling each type of subprocesses called MESingleTopTChannel, MESingleTopSChannel and MESingleTopTW, respectively.
• Processes involving the Higgs boson
• The MEPP2Higgs class implements the production of an s-channel Higgs boson via the $$gg, q\bar{q}\to h^0$$ processes. The matrix element correction for hard QCD radiation in the parton shower is included. The default instance is called MEHiggs.
• The MEPP2HiggsJet class implements the production of the Higgs boson in association with a hard jet via the $$gg\to gh^0, qg\to q h^0$$, $$\bar{q}g\to\bar{q}h^0$$ and $$q\bar{q}\to g h^0$$ processes. The default instance is called MEHiggsJet.
• The MEPP2WH class implements the production of the Higgs boson in association with the $$W^\pm$$, i.e. $$q\bar{q'}\to W^\pm h^0$$. The matrix element correction for hard QCD radiation in the parton shower is included. The default instance is called MEPP2WH.
• The MEPP2ZH class implements the production of the Higgs boson in association with the $$Z^0$$ boson, i.e. $$q\bar{q}\to Z^0h^0$$. The matrix element correction for hard QCD radiation in the parton shower is included. The default instance is called MEPP2ZH.
• The MEPP2HiggsVBF class implements the production of the Higgs boson via the vector-boson fusion process, i.e. $$qq\to qqh^0$$ via $$W^+W^-$$ and $$Z^0Z^0$$. The matrix element correction for hard QCD radiation in the parton shower is included. The default instance is called MEPP2HiggsVBF.
• The MEPP2QQHiggs class implements the production of the Higgs boson in association with a heavy quark-antiquark pair, i.e. $$q\bar{q},gg\to Q\bar{Q} h^0$$. Two instances are created by defaults, MEPP2bbbarH for production with a $$b\bar{b}$$ pair and MEPP2ttbarH for production with a $$t\bar{t}$$ pair.
• Matrix Elements for minimum-bias and underlying event physics.
• The MEQCD2to2Fast implements the same QCD $$2\to2$$ scattering processes as in the MEQCD2to2 class but uses hard-coded matrix elements rather than the helicity classes to increase the speed of the generation of secondary scattering processes.
• The MEMinBias class implements a soft-scattering process as part of the multiple scattering model of the underlying event and to enable the generation of minimum-bias events. An example of using it to generate min-bias events is supplied in LHC-MB.in.

In addition there are a small number of processes accurate to NLO in the POWHEG scheme. Examples of using these can be found in LHC-Powheg.in for the LHC and TVT-Powheg.in for the Tevatron.

• The MEqq2gZ2ffPowheg class implements the process $$q\bar{q}\to \gamma/Z^0\to f\bar{f}$$ including the NLO QCD corrections in the POWHEG scheme. The default instance is called PowhegMEqq2gZ2ff.
• The MEqq2W2ffPowheg class implements the process $$q\bar{q'}\to W^\pm\to f\bar{f'}$$ including the NLO QCD corrections in the POWHEG scheme. The default instance is called PowhegMEqq2W2ff.
• The MEPP2WHPowheg class implements the production of the Higgs boson in association with the $$W^\pm$$, i.e. $$q\bar{q'}\to W^\pm h^0$$, including the NLO QCD corrections in the POWHEG scheme. The default instance is called PowhegMEPP2WH.
• The MEPP2ZHPowheg class implements the production of the Higgs boson in association with the $$Z^0$$ boson, i.e. $$q\bar{q}\to Z^0h^0$$, including the NLO QCD corrections in the POWHEG scheme. The default instance is called PowhegMEPP2ZH.
• The MEPP2HiggsPowheg class implements the production of an s-channel Higgs boson via the $$gg, q\bar{q}\to h^0$$ processes including the NLO QCD corrections top the gluon-gluon process in the POWHEG scheme. The default instance is called PowhegMEHiggs.
• The MEPP2HiggsVBFPowheg class implements the production of the Higgs boson via the vector-boson fusion process, i.e. $$qq\to qqh^0$$ via $$W^+W^-$$ and $$Z^0Z^0$$, including the NLO QCD corrections in the POWHEG scheme. The default instance is called PowhegMEPP2HiggsVBF.
• The MEPP2VVPowheg class implements the production of pairs of massive electroweak gauge bosons, i.e. the processes $$q\bar{q}\to Z^0Z^0$$, $$q\bar{q}\to W^+W^-$$, $$q\bar{q'}\to W^\pm Z^0$$, including the NLO QCD corrections in the POWHEG scheme. The default instance is called PowhegMEPP2VV.
• Matrix Elements for lepton-lepton collisions. There are two example input files provided for lepton-lepton collisions. The first LEP.in is primarily intended to study physics at LEP collider energies and uses $$e^+e^-\to q\bar{q}$$ as the default scattering process with only d,u,s,c and b quarks produced. While the second illustrates the production of top quarks via $$e^+e^-\to t\bar{t}$$ for a future linear collider ILC.in. There are a number of matrix elements available.
• The MEee2gZ2qq class implements the process $$e^+e^-\to q\bar{q}$$ with an instance called MEee2gZ2qq and is the default scattering process for lepton-lepton collisions. This process also has the matrix-element correction for hard QCD radiation.
• The MEee2gZ2ll class implements the process $$e^+e^-\to \ell^+\ell^-$$ with an instance called MEee2gZ2ll.
• The MEee2HiggsVBF class implements the process $$e^+e^-\to e+ e^- h^0, \nu_e\bar{\nu}-e h^0$$ with an instance called MEee2HiggsVBF.
• The MEee2VV class implements the process $$e^+e^-\to W^+W^-, Z^0Z^0$$ with an instance called MEee2VV
• The MEee2ZH class implements the process $$e^+e^-\to Z^0 h^0$$ with an instance called MEee2ZH
• The MEee2VectorMeson class implements the process $$e^+e^-\to V^0$$ where $$V^0$$ is a spin-1 vector meson, e.g. $$J/\Psi, \Upsilon$$. By default no instance of this class is created and it is normally use to produce the $$\Upsilon(4S)$$ resonance to compare with B-factory measurements. An example of using it can be found in the Tests directory of the Herwig source in Tests/Rivet/BFactory-10.58.in.

In addition to the leading-order matrix elements we also provide two matrix-elements which are accurate to NLO in the POWHEG scheme. Examples of using these can be found in the LEP.in input file.

• The MEee2gZ2llPowheg class implements the process $$e^+e^-\to \ell^+\ell^-$$ including the NLO QED corrections for final-state radiation only. The default instance is called PowhegMEee2gZ2ll.
• The MEee2gZ2qqPowheg class implements the process $$e^+e^-\to q\bar{q}$$ including the QCD and final-state QED corrections. The default instance is called PowhegMEee2gZ2qq.
• Matrix Elements for lepton-hadron collisions. There are examples of using the matrix elements for lepton-hadron collisions in the DIS.in input file. For Deep Inelastic scattering (DIS) the QCD corrections in the POWHEG scheme are implemented in the base class for both the charged and neutral current processes the DIS.in input file also contains examples to include the NLO corrections in the POWHEG scheme. The matrix element correction for these processes is also available and switched on by default for the leading-order processes.
• The MENeutralCurrentDIS class implements the matrix elements for the neutral-current DIS process, i.e. $$q\ell^- \to q \ell^-, q\nu_\ell\to q\nu_\ell$$ and the charged conjugate processes. The default instance of this class is MEDISNC for leading-ordered processes and PowhegMEDISNC for next-to-leading ordered processes in the POWHEG scheme.
• The MEChargedCurrentDIS class implements the matrix elements for the charged-current DIS process, i.e. $$q \ell^- \to q' \nu_\ell, q \bar{\nu}_\ell \to q'\ell^-$$ and the charged conjugate processes. The default instance of this class is MEDISCC for leading-ordered processes and PowhegMEDISCC for next-to-leading ordered processes in the POWHEG scheme.
• Matrix Elements for gamma-gamma and gamma-hadron collisions. There are examples of using the matrix elements for $$\gamma\gamma$$ collisions in the case that the photons are radiated from incoming hadrons in LHC-GammaGamma.in and for incoming leptons in GammaGamma.in. There are two processes for $$\gamma\gamma$$ collisions.
• The MEGammaGamma2ff class implements $$\gamma\gamma\to f\bar{f}$$ and the default instance is MEgg2ff.
• The MEGammaGamma2WW class implements $$\gamma\gamma\to W^+W^-$$ and the default instance is MEgg2WW

In addition the MEGammaP2Jets with an instance called MEGammaP2Jets implements the processes $$\gamma q \to q g, \gamma \bar{q} \to \bar{q} g, \gamma g \to q \bar{q}$$ and can be used for $$\gamma p$$ collisions.