Parton Showers and Matrix Element Merging in Event

Parton Showers and Matrix Element Merging in Event Generatora Mini-Overview Tancredi. Carli@cern. ch 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 MC@NLO

ME Generators • 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. ) MC@NLO (on top of HERWIG)

Calculation of Hadron-Hadron Cross-Section Diverges for low PT 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 Outgoing partons radiate softer and softer partons Incoming partons radiate harder and harder partons For more complex reaction often not clear which subdiagram Should be treated as Hardest double counting

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

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

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

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

PS ME Merging • 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 WPS gives (real) parton shower phase space: correction factor WME/ WPS 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

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

CKKW-Merging 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. Jet production 2) Calculate ME weight for exclusive topologies up to ME cut 3) Make parton shower and veto parton shower above ME-cut Jet evolution reconstruct shower history “nodal” values for tree diagram Specifying k. T sequence for event

CKKW - ME-Weights and PS Veto 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 This procedure is included in SHERPA First implementation exist for PYTHIA++, HERWIG++ Procedure can be generalised to pp (Krauss 2002) One could start at dcut, but this would create a dip near dcut, so PS veto approach is better

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

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

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

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

MC@NLO Frixione, Webber 2002 In previous methods, the IR singularities in ME are cut and bias is corrected MC@NLO includes the virtual diagrams to cancel the IR singularities Problem: in NLO singularities cancel bin-by-bin, when shower is attached not possible Objectives: 1) 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 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 Event generator including benefits from NLO computations

MC@NLO For pp X Y: Introduce MC counterterms remove by hand non branching probability of Born term included in showers Shower generating functionals whose initial configuration Real MC Collinear counter-term is the 2 2 and 2 3 hard partons Born, virtual ME collinear remainder replaces formula d-function in Introduce MC counterterms subtraction method remove spurious NLO terms arising from the evolution of Born ME 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

Overview • Standard MC Generators: Pythia, Herwig, … 2 2 ME (at most) + parton showers+hadronisation LO only • Matrix Element Generators for specific processes: Unweighted events Acer. MC, Alpgen, Gr@ppa, Mad. Cup, Vecbos Up to 8 external particles • Matrix Element Generator for arbitrary processes: Amegic++/Sherpa, Comp. Hep, Grace, Mad. Event/Mad. Graph • ME at NLO precision with event generator MC@NLO, 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)

Literature • • F. Krauss, Matrix Elements and Parton Shower in Hadronic Interations, hep-ph/025283 T. Sjostrand, Monte Carlo Generator, hep-ph/0611247 S. Frixione, The Inclusion of Higher-Order QCD corrections into Parton Shower MC hep-ph/0408316 S. Mrenna and P. Richardson, Matching ME and PS with Herwig and Pythia, hep-ph/0312274