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M a t c h i n g a t L O a n d N L O I n t ro d u c t i o n t o Q C D - L e c t u re 4 Image Credits: istockphoto P. Skands (CERN) The Problem Lecture 2 : Matrix elements are correct When all jets are hard and there are no hierarchies

  1. M a t c h i n g a t L O a n d N L O I n t ro d u c t i o n t o Q C D - L e c t u re 4 Image Credits: istockphoto P. Skands (CERN)

  2. The Problem Lecture 2 : Matrix elements are correct When all jets are hard and there are no hierarchies (single-scale problem = small corner of phase space, but an important one!) But they are unpredictive for strongly ordered emissions Lecture 3 : Parton Showers are correct When all emissions are (successively) strongly ordered (= dominant QCD structures) But they are unpredictive for hard jets Often too soft (but not guaranteed! Can also err by being too hard!) ME-PS matching → ONE calculation to rule them all QCD Lecture IV P . Skands 2

  3. Example: . Born + Shower 2 2 + + … Shower Approximation to Born + 1 QCD Lecture IV P . Skands 3

  4. Example: . Born + Shower 2 2 + + … Shower Approximation to Born + 1 Born + 1 @ LO 2 + QCD Lecture IV P . Skands 3

  5. Example: . Born + Shower 2  2 s ik ✓ s ij 1 ◆� 1 + s jk + … + = g 2 s 2 C F + s ij s jk s IK s jk s ij Born + 1 @ LO 2  2 s ik ✓ s ij ◆� 1 + s jk = g 2 s 2 C F + + 2 s ij s jk s IK s jk s ij QCD Lecture IV P . Skands 4

  6. Example: . Born + Shower 2  2 s ik ✓ s ij 1 ◆� 1 + s jk + … + = g 2 s 2 C F + s ij s jk s IK s jk s ij Born + 1 @ LO 2  2 s ik ✓ s ij ◆� 1 + s jk = g 2 s 2 C F + + 2 s ij s jk s IK s jk s ij QCD Total Overkill to add these two. All I really need is just that +2 … Lecture IV P . Skands 4

  7. Adding Calculations Born × Shower X+1 @ LO (see lecture 3) (with p T cutoff, see lecture 2) X (2) X+1 (2) … X+1 (2) … X+1 (1) X+2 (1) X+3 (1) X+1 (1) X+2 (1) X+3 (1) X (1) … … X+1 (0) X+2 (0) X+3 (0) X+1 (0) X+2 (0) X+3 (0) Born … … Fixed-Order ME above p T cut … Fixed-Order Matrix Element … & nothing below … Shower Approximation QCD Lecture IV P . Skands 5

  8. Adding Calculations Born × Shower X+1 @ LO × Shower (see lecture 3) (with p T cutoff, see lecture 2) X (2) X+1 (2) … X+1 (2) … X+1 (1) X+2 (1) X+3 (1) X+1 (1) X+2 (1) X+3 (1) X (1) … … X+1 (0) X+2 (0) X+3 (0) X+1 (0) X+2 (0) X+3 (0) Born … … Fixed-Order ME above p T cut … Fixed-Order Matrix Element … & nothing below Shower approximation above p T cut … Shower Approximation … QCD & nothing below Lecture IV P . Skands 6

  9. → Double Counting Born × Shower + (X+1) × shower X (2) X+1 (2) … Double Counting of X+1 (1) X+2 (1) X+3 (1) X (1) … Worse than useless terms present in both expansions X+1 (0) X+2 (0) X+3 (0) Born … … Fixed-Order Matrix Element Double counting above p T cut … & shower approximation below … Shower Approximation QCD Lecture IV P . Skands 7

  10. Interpretation ► A (Complete Idiot’s) Solution – Combine 1. [X] ME + showering Run generator for X (+ shower) 2. [X + 1 jet] ME + showering Run generator for X+1 (+ shower) 3. … Run generator for … (+ shower) Combine everything into one sample ► Doesn’t work • [X] + shower is inclusive • [X+1] + shower is also inclusive ≠ What you What you want get Overlapping “bins” One sample QCD Lecture IV P . Skands 8

  11. Cures QCD Lecture IV P . Skands 9

  12. Cures X X (2) … +1 (2) Tree-Level Matrix Elements X X X X (1) … +1 (1) +2 (1) +3 (1) X X (2) … +1 (2) PHASE-SPACE SLICING (a.k.a. CKKW, MLM, …) X X X … Born +1 (0) +2 (0) +3 (0) X X X X (1) … +1 (1) +2 (1) +3 (1) UNITARITY (a.k.a. merging, PYTHIA, X X X VINCIA, …) … Born +1 (0) +2 (0) +3 (0) QCD Lecture IV P . Skands 9

  13. Cures X X (2) … +1 (2) Tree-Level Matrix Elements X X X X (1) … +1 (1) +2 (1) +3 (1) X X (2) … +1 (2) PHASE-SPACE SLICING (a.k.a. CKKW, MLM, …) X X X … Born +1 (0) +2 (0) +3 (0) X X X X (1) … +1 (1) +2 (1) +3 (1) UNITARITY (a.k.a. merging, PYTHIA, X X X VINCIA, …) … Born +1 (0) +2 (0) +3 (0) NLO Matrix Elements X X (2) … +1 (2) SUBTRACTION (a.k.a. MC@NLO) X X X X (1) … +1 (1) +2 (1) +3 (1) UNITARITY + SUBTRACTION (a.k.a. POWHEG, VINCIA) X X X … Born +1 (0) +2 (0) +3 (0) QCD Lecture IV P . Skands 9

  14. Cures X X (2) … +1 (2) Tree-Level Matrix Elements X X X X (1) … +1 (1) +2 (1) +3 (1) X X (2) … +1 (2) PHASE-SPACE SLICING (a.k.a. CKKW, MLM, …) X X X … Born +1 (0) +2 (0) +3 (0) X X X X (1) … +1 (1) +2 (1) +3 (1) UNITARITY (a.k.a. merging, PYTHIA, X X X VINCIA, …) … Born +1 (0) +2 (0) +3 (0) NLO Matrix Elements X X (2) … +1 (2) SUBTRACTION (a.k.a. MC@NLO) X X X X (1) … +1 (1) +2 (1) +3 (1) UNITARITY + SUBTRACTION (a.k.a. POWHEG, VINCIA) X X X … Born +1 (0) +2 (0) +3 (0) + WORK IN PROGRESS … X (2) X+1 (2) … NLO + multileg tree-level matrix elements X (1) X+1 (1) X+2 (1) X+3 (1) … Born X+1 (0) X+2 (0) X+3 (0) … NLO multileg matching X (2) X+1 (2) … X (1) X+1 (1) X+2 (1) X+3 (1) … X (2) X+1 (2) … Matching at NNLO QCD X (1) X+1 (1) X+2 (1) X+3 (1) … Born X+1 (0) X+2 (0) X+3 (0) … Lecture IV Born X+1 (0) X+2 (0) X+3 (0) … P . Skands 9

  15. Cures X X (2) … +1 (2) Tree-Level Matrix Elements X X X X (1) … +1 (1) +2 (1) +3 (1) X X (2) … +1 (2) PHASE-SPACE SLICING (a.k.a. CKKW, MLM, …) X X X … Born +1 (0) +2 (0) +3 (0) X X X X (1) … +1 (1) +2 (1) +3 (1) UNITARITY (a.k.a. merging, PYTHIA, X X X VINCIA, …) … Born +1 (0) +2 (0) +3 (0) NLO Matrix Elements X X (2) … +1 (2) SUBTRACTION (a.k.a. MC@NLO) X X X X (1) … +1 (1) +2 (1) +3 (1) UNITARITY + SUBTRACTION (a.k.a. POWHEG, VINCIA) X X X … Born +1 (0) +2 (0) +3 (0) + WORK IN PROGRESS … X (2) X+1 (2) … NLO + multileg tree-level matrix elements X (1) X+1 (1) X+2 (1) X+3 (1) … Born X+1 (0) X+2 (0) X+3 (0) … NLO multileg matching X (2) X+1 (2) … X (1) X+1 (1) X+2 (1) X+3 (1) … X (2) X+1 (2) … Matching at NNLO QCD X (1) X+1 (1) X+2 (1) X+3 (1) … Born X+1 (0) X+2 (0) X+3 (0) … Lecture IV Born X+1 (0) X+2 (0) X+3 (0) … P . Skands 9

  16. X X (2) … +1 (2) X X X X (1) … +1 (1) +2 (1) +3 (1) X X X Born … +1 (0) +2 (0) +3 (0) Phase-Space Slicing Matching to Tree-Level Matrix Elements A.K.A. CKKW, CKKW-L, MLM 10

  17. Phase Space Slicing (with “matching scale”) Born × Shower X+1 @ LO × Shower + shower veto above p T with 1 jet above p T X (2) X+1 (2) … X+1 (2) … X+1 (1) X+2 (1) X+3 (1) X+1 (1) X+2 (1) X+3 (1) X (1) … … X+1 (0) X+2 (0) X+3 (0) X+1 (0) X+2 (0) X+3 (0) Born … … Fixed-Order ME above p T cut … Fixed-Order Matrix Element … & nothing below … Shower Approximation QCD Lecture IV P . Skands 11

  18. Phase Space Slicing (with “matching scale”) Born × Shower + X+1 @ LO × Shower + shower veto above p T with 1 jet above p T X (2) X+1 (2) … X+1 now correct in X+1 (1) X+2 (1) X+3 (1) both soft and hard X (1) … limits X+1 (0) X+2 (0) X+3 (0) Born … Fixed-Order ME above p T cut … Fixed-Order Matrix Element … & nothing below Fixed-Order ME above p T cut … Shower Approximation … QCD & Shower Approximation below Lecture IV P . Skands 12

  19. Multi-Leg Slicing (a.k.a. CKKW or MLM matching) CKKW: Catani, Krauss, Kuhn, Webber, JHEP 0111:063,2001. Keep going MLM: Michelangelo L Mangano Veto all shower emissions above “matching scale” Except for the highest-multiplicity matrix element (not competing with anyone) X (2) X+1 (2) … Multileg Precision: Tree-level LO: when all jets hard Still LL: for soft emissions X+1 (1) X+2 (1) X+3 (1) matching: X (1) … X+1 (0) X+2 (0) X+3 (0) Born … QCD Lecture IV P . Skands 13

  20. Classic Example mcplots.cern.ch W + Jets Number of Jets Number of jets in pp → W+X at the LHC From 0 (W inclusive) to W+3 jets With Matching LHC 7 TeV PYTHIA includes Without Matching W+Jets matching up to W+1 jet E Tj > 20 GeV | η j | < 2.8 + shower With ALPGEN, also the LO matrix elements for RATIO 2 and 3 jets are included But Normalization still only LO QCD Lecture IV P . Skands 14

  21. Classic Example mcplots.cern.ch W + Jets Number of Jets Number of jets in pp → W+X at the LHC From 0 (W inclusive) to W+3 jets With Matching LHC 7 TeV PYTHIA includes Without Matching W+Jets matching up to W+1 jet E Tj > 20 GeV | η j | < 2.8 + shower With ALPGEN, also the LO matrix elements for RATIO 2 and 3 jets are included But Normalization still only LO QCD Lecture IV P . Skands 14

  22. Slicing: Some Subtleties Choice of slicing scale (=matching scale) Fixed order must still be reliable when regulated with this scale → matching scale should never be chosen more than ~ one order of magnitude below hard scale. Precision still “only” Leading Order Choice of Renormalization Scale We already saw this can be very important (and tricky) in multi-scale problems. Caution advised (see also supplementary slides & lecture notes) QCD Lecture IV P . Skands 15

  23. Choice of Matching Scale Reminder: in perturbative region, QCD is approximately scale invariant Low Matching Scale → A scale of 20 GeV for a W boson becomes 40 GeV for something weighing 100 2M W, etc … (+ adjust for C A /C F if g-initiated) 75 → The matching scale should be written as 50 a ratio (Bjorken scaling) Using a too low matching scale → 25 everything just becomes highest ME 0 Caveat emptor: showers generally do not Born (exc) + 1 + 2 (inc) QCD include helicity correlations Lecture IV P . Skands 16

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