Introduction to jet-parton matching in MG/ME

The aim of any parton-jets matching procedure is mainly to avoid overlapping between phase-space descriptions given by matrix-element generators and showering/hadronization softwares in multi-jets process simulation. The motivation for using both at the same time is the following:

• The Parton Shower (PS) Monte Carlo programs such as Pythia and Herwig describe parton radiation as successive parton emissions using Markov chain techniques based on Sudakov form factors. This description is formally correct only in the limit of soft and collinear emissions, but has been shown to give a good description of much data also relatively far away from this limit. However, for the production of hard and widely separated QCD radiation jets, this description breaks down due to the lack of subleading terms and interference. For that case, it is necessary to use the full tree-level amplitudes for the heavy particle production plus additional hard partons.

• The Matrix Element (ME) description diverges as partons become soft or collinear, while the parton shower description breaks down when partons become hard and widely separated.

We can distinguish two different philosophies/method types: either based on shower veto and therefore a event reweighting (CKKW method) or events rejection. The latter is the method adopted in the MLM-based schemes. Note that in the CKKW case, partons are clustered in jets with the $K_{T}$ algorithm while the original MLM method uses a cone algorithm and minimum $P_{T}$ cut. In MadGraph/MadEvent, there are currently three matching schemes implemented, all based on MLM method. They are called cone- and $K_{T}$-jet MLM and Shower-$K_{T}$ respectively. In all cases the parton shower generator is Pythia.

The dependence to the parton-shower evolution variable is here important. The MLM schemes can be used with both "old" virtuality-ordered showers ( MSTP(81)=0 or 1 in Pythia) and "new" $P_{T}$-ordered showers (( MSTP(81)=20 or 21 in Pythia)), whereas the Shower-$K_{T}$ has been designed only for the latter.

• Cone- or $K_{T}$-jet MLM: The final-state partons in an MG/ME event are clustered according to the $K_{T}$-jet algorithm to find the "equivalent parton shower history of the event. The Feynman diagram information from MadGraph is used to allow only clusterings that correspond to diagrams existing in the generated matrix element. For the cone jet algorithm, a minimum $P_T$ and $\Delta R$ must be defined for all partons. For the $K_T$ scheme, the smallest $K_{T}$ value is restricted to be above the cutoff scale "xqcut". In order to closely mimic the behaviour of the parton shower, the $K_{T}$ value for each clustering vertex corresponding to a QCD emission is used as renormalization scale for $\alpha_{s}$ in that vertex. As factorization scale, as well as renormalization scale for the central hard $2\to1$ or $2\to2$ process, the transverse mass $m_{T}^{2 = P_{T}}2 + m^{2$ of the particle(s) produced in the central process is used. This event is then passed to Pythia for parton showering. After showering, but before hadronization and decays, the final-state partons are clustered into jets; for the cone jet MLM scheme using cone jets, with minimum $P_T > P_T}{ME}$ and $\Delta R=\Delta R^{{ME}$, and for the $K_{T}$-jet scheme using $K_{T}$ jets with a cutoff scale $Qcut > xqcut$. These jets are then compared to the original partons from the matrix element event. A jet is considered to be matched to the closest parton if, for the cone jet scheme, the jet is within 1.5$\Delta R$ from the parton, and for the $K_{T}$-jet scheme, if the jet measure $K_{T}(parton,jet)$ is smaller than the cutoff $Qcut$. The event is rejected unless each jet is matched to a parton, except for the highest multiplicity sample, where extra jets are allowed below the $P_T$ or $K_{T}$ scale (for the respective schemes) of the softest matrix element parton in the event. These matching schemes can be used with both the old (vituality-ordered) and the new ($\pt$-ordered) shower implementations of \pythia.}

• Shower-$K_{T}$: In this scheme, events are generated by MG/ME as described above, including the reweighting of $\alpha_{s}$. The event is then passed to Pythia and showered using the $P_{T}$-ordered showers. In this case, Pythia reports the scale of the first (hardest) emission in the shower, $Q_{hard}$. For events from lower-multiplicity samples, the event is rejected if $Q_{hard}$ is above the matching scale $Qcut$, while events from the highest multiplicity sample are rejected if $Q_{hard} > Q^{{ME}{low}$, the scale of the softest matrix element parton in the event. This matching scheme is simpler and yet effectively mimics the workings of the $K{T}$-jet MLM scheme. However, it allows for the matching scale $Qcut$ to be set equal to the matrix element cutoff scale $xqcut$, and it more directly samples the Sudakov form factor used in the shower. Furthermore, the treatment of the highest multiplicity sample more closely mimics that used in the CKKW matching scheme.}

Practical aspects

For a discussion of the parameters used in the MadEvent-Pythia matching, please see the page Matching of jets between MadEvent and Pythia.

We now give a concrete example of how to generate properly a matched sample using MadGraph/MadEvent/Pythia and $K_{T}$-MLM scheme. In this example, we show how it is possible to produce a matched sample of tt + 0, 1, 2 jets (inclusively).

The first step consists in modifying the proc card.dat file, which will contain all the information about the process itself. In our case we will use (the headers and footers have been removed in order to save place):

pp>tt~ @0 # First SubProcess
QCD=99 # Max QCD couplings
QED=0 # Max QED couplings
end_coup

pp>tt~j @1 # Second SubProcess
QCD=99 # Max QCD couplings
QED=0 # Max QED couplings
end_coup

pp>tt~jj @2 # Third SubProcess
QCD=99 # Max QCD couplings
QED=0 # Max QED couplings
end_coup
done

• Remark 1: The number of QED vertices has to be reduced as much as possible: the reason is twofold: first a parton emitted by a QED vertex cannot be renormalized with $\alpha_{s}$. Second, in the parton shower there is no QED emission of parton, and the matching procedure is supposed to be precisely merge similar things at the point of view of the content.
• Remark 2: the maximal number of extra-partons is strongly related to the process and model used, because of the practical limitation of the number of diagrams in Madgraph: While in the Standard Model we can easily reach 4 extra-partons for W/Z+jets, in the MSSM this number is limited to 2 for processes like $\tilde{g}\tilde{g}$+jets.

After generating the processes, the run card.dat can be edited as well as the pythia card.dat. Those two files will contains everything needed to perform the matching procedure.

• At the level of the run_card, the most important parameters are ickkw, ptj, etaj, xqcut, drjj.
  • ickkw has to be set to 1 to perform MLM-type matching.
  • xqcut defines the minimal distance in the phase space allowed between extra partons (u,d,s,c, and also b if maxjetflavor=5 in the run_card). For t t~ a value around 15 or 20 is reasonable.
  • drjj is the distance in the eta-phi plan between partons. As the whole matching procedure uses the $K_{T}$ measure to control this parameter, the drjj can be set to a very low value (like 0.001) to not influence the xqcut. For MG/ME v. 4.4.16 and higher, drjj can be set to 0 if xqcut is > 0.
  • etaj should be set to 5 for the LHC (which is the maximal rapidity used in the matching procedure ).
  • The xqcut definition is of $K_{T}$ type, it means then also related to $P_{T}$. Therefore, in order to optimize the speed of even generation by restricting the phase-space on which calculation is allowed, ptj can be set equal to xqcut. This is not necessary for MG/ME v. 4.4.16 and higher, where ptj can be set to 0.

• For the pythia step, where the matching procedure takes effectively place the card can be written as follows:

  !...Parton showering on or off
  MSTP(61)=1
  MSTP(71)=1
  !...Fragmentation/hadronization on or off
  MSTJ(1)=1
  !...Multiple interactions on or off
  MSTP(81)=20
  !...Cutoff in jet measure for matching
  QCUT = 30
  

The result is a file in .hep format (STDHEP) which is inclusive in the multiplicities of extra jet radiation. If the user want to use the Shower-$K_{T}$ instead of the $K_{T}$-MLM scheme, the parameter "showerkt=T" has to be included in the pythia_card.

Suggested scale choices

• For W or Z boson production, suggested xqcut scale is 10 !GeV with QCUT=15 !GeV for virtuality-ordered Pythia showers, or 30 !GeV for $P_T$-ordered showers with the Shower-$K_T$ scheme.
• For t t~ production, suggested xqcut scale is 20 !GeV with QCUT=30 !GeV for virtuality-ordered Pythia showers, or 80 !GeV for $P_T$-ordered showers with the Shower-$K_T$ scheme.
• For 600 !GeV SUSY particle pair production, suggested xqcut scale is 30 !GeV with QCUT=40 !GeV for virtuality-ordered Pythia showers, or 100 !GeV for $P_T$-ordered showers with the Shower-$K_T$ scheme.

Matching in BSM processes

For many processes involving chains of heavy particles (like for example gluinos and squarks), a second problem of double counting appears. Let's take for example the production of a pair of squarks + 0,1,2 partons: in the case where one of two extra-partons are present, the presence of a gluino resonance can contribute largely. However diagram with gluinos decaying each into a squark and a jet are already present in the production gluino + extra-partons at the ME-level, with their decay into squark and jet at the PS shower level. To get rid of this problem a treatment of events containing these resonances has been set up: at the Pythia level, if the event is flagged as containing a resonance (with a given PDG code), it is automatically rejected. The syntax is very simple and limited at some lines in the pythia_card.dat:

EXCRES= PDG_code_of_the_resonance_to_be_excluded

One line per resonance to be excluded can be added.

How to check that everything worked?

See the MatchChecker page.

Further remarks

• In the standard version of Pythia installed in the MG/ME package, the cross-section after matching (equivalent to (the cross-section before matching)*(Sudakov rejection)), can be found at the very end of the pythia.log file as well as the matching efficiency.
• The cross-sections delivers are to be taken with some care. It is of course a onky a tree-level calculation, so what should be trusted are the shapes of the distributions, not their normalization. A simple example is the difference of cross-section when different showers parameterization are used.

Where to find more documentation?

• QCD radiation in the production of heavy colored particles at the LHC. Johan Alwall, Simon de Visscher, Fabio Maltoni . Oct 2008. 26pp. Published in JHEP 0902:017,2009. ​http://arxiv.org/abs/0810.5350.
• Comparative study of various algorithms for the merging of parton showers and matrix elements in hadronic collisions. J. Alwall et al. SLAC-PUB-12604, CERN-PH-TH-2007-066, LU-TP-07-13, KA-TP-06-2007, DCPT-07-62, IPPP-07-31, Jun 2007. 44pp. Published in Eur.Phys.J.C53:473-500,2008. ​http://arxiv.org/abs/0706.2569.
• MadGraph/MadEvent v4: The New Web Generation. Johan Alwall (SLAC) , Pavel Demin, Simon de Visscher, Rikkert Frederix, Michel Herquet, Fabio Maltoni (Louvain U., CP3 & IBA, Louvain-la-Neuve) , Tilman Plehn (Edinburgh U.) , David L. Rainwater (Rochester U.) , Tim Stelzer (Illinois U., Urbana) . SLAC-PUB-12603, CP3-07-17, Jun 2007. 38pp. Published in JHEP 0709:028,2007.
• QCD matrix elements + parton showers. S. Catani (CERN) , F. Krauss (Cambridge U.) , R. Kuhn (Dresden, Tech. U. & Dresden, Max Planck Inst.) , B.R. Webber (Cambridge U.) . CERN-TH-2000-367, CAVENDISH-HEP-00-03, Sep 2001. 21pp. Published in JHEP 0111:063,2001. ​http://arxiv.org/abs/0706.2334.

-- Main.SimonDeVisscher - 02 Mar 2009

-- Main.JohanAlwall - 2011-03-16