Parton Showers and Matrix Element Merging in Event Generator- a Mini-Overview Introduction to ME+PS Branching and Sudakov factor (no branching) Matching ME 2  3 + PS –CKKW and MLM Mini-Reminder: NLO calculation and subtraction method

Since mid 1980 Pythia and Herwig (and Ariadne) have been widely used in most particle physics experiments They provide full topology of final state particles based on: Born matrix element + Parton shower (leading log approx.) for higher order for radiation in strong and electromagnetic interaction + hadronisation (Lund strong and cluster model) Late 80/early 90: Procedure to work out merging ME+PS, i.e. ME is used for emission of additional jet besides Born process (Sjostrand, Seymour, Lonnblad) Starting from 2001/2002: A lot of activity started (…triggered by LHC challenge: large phase space!) 1) General Algorithm to merge ME(2  n)+PS in leading order (Mangano, Krauss, Catani, Webber et al.)  included in SHERPA, ALPGEN, HERWIG++ 2) Match NLO ME + Parton showers (Frixione, Webber, Nason et al.)  (on top of HERWIG) ME Generators

Calculation of Hadron-Hadron Cross-Section  Diverges for low P T  0 The calculation of exact matrix elements is difficult (loops, divergences, cancellations between large positive/negative numbers)

The Parton Shower Approximation Hard 2  2 process calculation has all (external leg) partons on mass shell However, partons can be off-shell for short times (uncertainty principle) close to the hard interaction Incoming partons radiate harder and harder partons Outgoing partons radiate softer and softer partons For more complex reaction often not clear which subdiagram Should be treated as Hardest  double counting

Final State Parton shower e.g.: Halzen&Martin chap 11 In the qg-collinear limit x 2  1

Altarelli-Parisi Splitting Function Iteration over branchings gives final state Parton shower

The Sudakov Form Factor P to branch first time= P to branch times P that no branch before Sudakov Sudakov form factor approximates the virtual loop corrections valid in collinear&soft limit x,Q 2 small

Details of Parton shower more complex, e.g. coherence, angular ordering Rate for one emission: Rate for n emissions: Parton shower include all corrections of type (better than analytical leading log)

Traditional ME/PS merging, e.g. in Pythia and Herwig details different in all MC: generate phase space with PS, correct first or hardest emission with ME probability If W PS gives (real) parton shower phase space: correction factor W ME/ W PS In this way effectively the splitting functions are replaced by the ME The reweighting only works, if in the full phase space of gluon emission This relation is not valid for higher parton configurations  reweighting procedure has to be satisfied PS ME Merging

Summary - …so far Parton showers include soft and collinear radiation that is logarithmically enhanced (non-singular contributions are ignored)  not enough gluons are emitted that have high energy and large angle from the shower initiator Matrix elements gives a good description of specific parton topologies where the partons are energetic and well separated, They include the interference between amplitudes with same external partons However, in the soft and collinear limit, they neglect interference between multiple gluon emissions, e.g. angular ordering

Jet Rates in NLL Accuracy No emission from each quark line No emission from quark line No emission from internal lines Branching at d 2 Two possible histories q and qbar can radiate Cluster partons to jets using K T -algorithm Stop at point where 2-jets (d 1 ), 3-jets (d 3 ) are resolved Sum over all possible branchings!

CKKW-Merging reconstruct shower history “nodal” values for tree diagram Specifying k T sequence for event Catani, Krauss, Kuhn, Webber (2001) General scheme to merge parton showers with ME 2->n 1) Make exclusive ME topologies, exactly 2-jets, 3-jets etc. 2) Calculate ME weight for exclusive topologies up to ME cut 3) Make parton shower and veto parton shower above ME-cut Jet production Jet evolution

CKKW - ME-Weights and PS Veto One could start at dcut, but this would create a dip near dcut, so PS veto approach is better This procedure is included in SHERPA First implementation exist for PYTHIA++, HERWIG++ Procedure can be generalised to pp (Krauss 2002) Had we known the branching tree we should have computed the MEs like that PS would not emit partons in addition to those in ME (exclusive) Avoid duble counting, well separated partons already done via ME

CKKW Result ME+PS Pythia default (dashed) ME 2  2 ME 2  3 ME 2  4 ME 2  5 ME 2  6 of parton i Note that the CKKW work up to NLL accuracy (for hadron collisions, no formal proof) When using CKKW, always make sure that d ini dependence is small

MLM Mangano (2002) 1) Generate hard parton configuration for given n=N part with ME, imposing 2) Define tree branching structure using K T -algo allowing only pairing consistent with color flow 3) Compute  s at the nodal values, but do not apply Sudakov factors 4) Shower the hard event without any veto using Herwig/Pythia when done, find N jet jet with cone algorithm with if N part <N jet reject event 5) Matched jets to hard partons using Only keep events, if each hard parton is uniquely contained in jets Events with N part <N jet are rejected except for highest multiplicities 6) Define exclusive N-jet sample by requiring N part =N jet 7) After matching, combine exclusive samples to one inclusive sample Used in Alpgen Hard parton Shower parton N part =N jet Event kept N part =N jet =3 but Nmatch=2 event rejected N part <N jet reject for excl. sample keep for incl. sample soft double counting collinear double counting This is equivalent to tSudakov reweighting in CKKW (external lines) Prevent parton shower harder than any emission by ME using cone algo:

Reminder: Next-To-Leading-Order calculations Born:First-Order: RealFirst Order: Virtual cancel each other (KNL-theorem), if infra-red singularities One can show that for any observable where the NLO prediction is: Loop diagram Real and virtual contributions can be regularised by introducing integral in d=4-2  dim. In this case: where: (infra-red safeness)

Subtraction Method Add and subtract locally a counter-term with same point-wise singular behaviour as R(x): Ellis, Ross, Terrano (1981) Since regularised Let us look at the real contribution: By construction this integral is finite Add and subtract counter-term The only divergent term has B&V kinematics and gets cancels against  s B/2  term of virtual contribution  cancellation independent of Observable

In previous methods, the IR singularities in ME are cut and bias is corrected includes the virtual diagrams to cancel the IR singularities Frixione, Webber ) Total rates are accurate in NLO (normalisation is meaningful, in contrast to LO MC) 2) Hard emissions are treated as in NLO computation (up to 2  3) 3) Soft/collinear emissions are treated as in MC, i.e. using PS 4) Smooth matching between hard and soft/collinear emissions 5) output set of event using standard hadronisation models Problem: in NLO singularities cancel bin-by-bin, when shower is attached not possible Event generator including benefits from NLO computations Objectives: Basic Scheme: 1) Calculate NLO ME for n-body process using subtraction method (n+1 real, n virtual+Born) 2) Calculate analytically how first shower emission off n-body topology populates n+1 phase space 3) Subtract the shower expression from the real n+1 ME, consider rest as n-body 4) Add shower to n and n+1 events

Real MC Born, virtual ME collinear remainder Collinear counter-term Introduce MC counterterms remove spurious NLO terms arising from the evolution of Born ME Shower generating functionals whose initial configuration is the 2  2 and 2  3 hard partons replaces formula  function in subtraction method Introduce MC counterterms remove by hand non branching probability of Born term included in showers For pp  X Y: Counter terms are constructed by hand to reproduce behaviour of collinear singularities, they locally match the singular behaviour of real ME. They are specific to MC implementation  So far, only HERWIG

Standard MC Generators: Pythia, Herwig,… 2  2 ME (at most) + parton showers+hadronisation Matrix Element Generators for specific processes: AcerMC, Alpgen, MadCup, Vecbos Matrix Element Generator for arbitrary processes: Amegic++/Sherpa, CompHep, Grace, MadEvent/MadGraph ME at NLO precision with event generator POWHEG Possible future development: Automatic matching of NLO ME + NLO parton showers Automatised NLO calculation plus matching Present theory bottleneck to go to NNLO or NLO for more then 2  3 (two-loop diagrams) Overview LO only Unweighted events Up to 8 external particles

F. Krauss, Matrix Elements and Parton Shower in Hadronic Interations, hep-ph/ T. Sjostrand, Monte Carlo Generator, hep-ph/ S. Frixione, The Inclusion of Higher-Order QCD corrections into Parton Shower MC hep-ph/ S. Mrenna and P. Richardson, Matching ME and PS with Herwig and Pythia, hep-ph/ Literature