Parton Showers and Matching/Merging Lecture 2 of 2: Matching/Merging & Non-Perturbative Corrections (Hadron Decays) Hadronisation Parton Shower Matching & Merging Hard process Peter Skands (Monash University) Feynrules/Madgraph School, Hefei 2018
SHOWERS VS MATRIX ELEMENTS ๏ Showers. Nice to have all-orders solution • But only exact in singular (soft & collinear) limits • → gets bulk of bremsstrahlung corrections right, but no precision for hard wide-angle radiation: visible, extra jets • … which is exactly where fixed-order (ME) calculations work! = ? So combine them! F & F+1 @ LO × LL F @ LO × LL F+1 @ LO × LL � (2) � (2) . . . 2 � (2) � (2) � (2) � (2) . . . . . . 2 2 0 1 0 1 0 1 = + ` (loops) ` (loops) ` (loops) � (1) � (1) � (1) . . . � (1) � (1) � (1) � (1) � (1) � (1) 1 . . . . . . 1 1 0 1 2 0 1 2 0 1 2 � (0) � (0) � (0) � (0) . . . � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) 0 . . . . . . 0 0 0 1 2 3 0 1 2 3 0 1 2 3 . . . 0 1 2 3 . . . . . . 0 1 2 3 0 1 2 3 k (legs) k (legs) k (legs) Matching See also: PS, Introduction to QCD , TASI 2012, arXiv:1207.2389 � 2 Peter Skands Monash University
HOW NOT TO DO IT … IN MORE DETAIL ► 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 3 � Peter Skands Monash University
EXAMPLE: . Born + Shower What the first-order shower expansion gives you 2 2 + + … Shower Approximation Born + 1 @ LO to Born + 1 2 + What you get from first-order (LO) madgraph 4 � Peter Skands Monash University
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 Example of shower kernel (here, used an “antenna function” for coherent gluon emission from a quark pair) 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 Example of matrix element; what MG would give you Total Overkill to add these two. All we really need is just that +2 … 5 � Peter Skands Monash University
1. MATRIX-ELEMENT CORRECTIONS Bengtsson, Sjöstrand, ๏ Exploit freedom to choose non-singular terms PLB 185 (1987) 435 • Modify parton shower to use process-dependent radiation functions for first emission → absorb real correction (suppressing | M n +1 | 2 → P 0 ( z ) Parton Shower P ( z ) = P ( z ) α s and P Jacobian i P i ( z ) /Q 2 Q 2 Q 2 Q 2 i | M n | 2 factors) | {z } MEC Process-dependent MEC → P’ different for each process ๏ • Done in PYTHIA for all SM decays and many BSM ones Norrbin, Sjöstrand, NPB 603 (2001) 297 Based on systematic classification of spin/colour structures ๏ Also used to account for mass effects, and for a few 2 → 2 procs ๏ ๏ Difficult to generalise beyond one emission • Parton-shower expansions complicated & can have “dead zones” • Achieved in VINCIA (by devising showers that have simple expansions) Giele, Kosower, Skands, PRD 84 (2011) 054003 • Only recently done for hadron collisions Fischer et al, arXiv:1605.06142 6 � Peter Skands Monash University
MECS WITH LOOPS: POWHEG Acronym stands for: Po sitive W eight H ardest E mission G enerator. Nason, JHEP 0411 (2004) 040 Start at Born level Loops Frixione, Nason, Oleari JHEP 0711 (2007) 070 + POWHEG Box JHEP 1006 (2010) 043 | M F | 2 +2 Note: still LO for X+1 Generate “shower” emission Shower for X+2, … | M F +1 | 2 LL X a i | M F | 2 +1 ∼ X i ∈ ant r e +0 w Correct to Matrix Element ∈ o h s | M F +1 | 2 n P a i | M F | 2 a i a i → +0 +1 +2 +3 Legs o t r a p y ๏ Method is widely applied/available, can be used r P | | a Unitarity of Shower n i with PYTHIA, HERWIG, SHERPA d r Z o : Virtual = − Real t a ๏ Subtlety 1: Connecting with parton shower e p Z e R • Truncated Showers & Vetoed Showers Correct to Matrix Element Z | M F | 2 → | M F | 2 + 2Re[ M 1 ๏ Subtlety 2: Avoiding (over)exponentiation of F M 0 F ] + Real hard radiation • Controlled by “hFact” parameter (POWHEG) 7 � Peter Skands Monash University
2 : SLICING (MLM & CKKW-L) First emission : “the HERWIG correction” Use the fact that the angular-ordered HERWIG parton shower has a “dead zone” for hard wide-angle radiation (Seymour, 1995) F @ LO × LL-Soft (H ERWIG Shower) F+1 @ LO × LL (H ERWIG Corrections) F @ LO 1 × LL (H ERWIG Matched) ! � (2) � (2) � (2) � (2) � (2) � (2) . . . . . . . . . 2 2 2 0 1 0 1 0 1 + = ` (loops) ` (loops) ` (loops) � (1) � (1) � (1) � (1) � (1) � (1) � (1) � (1) � (1) . . . . . . . . . 1 1 1 ! 0 1 2 0 1 2 0 1 2 � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) . . . . . . . . . 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 . . . . . . . . . 0 1 2 3 0 1 2 3 0 1 2 3 k (legs) k (legs) k (legs) Many emissions : the MLM & CKKW-L prescriptions F @ LO × LL-Soft (excl) F+1 @ LO × LL-Soft (excl) F+2 @ LO × LL (incl) F @ LO 2 × LL (MLM & (L)-CKKW) � (2) � (2) � (2) � (2) . . . . . . . . . . . . 2 2 2 2 0 0 0 0 + + = ` (loops) ` (loops) ` (loops) ` (loops) � (1) � (1) � (1) � (1) � (1) � (1) � (1) � (1) . . . . . . . . . . . . 1 1 1 1 0 1 0 1 0 1 0 1 � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) � (0) 0 0 0 0 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 k (legs) k (legs) k (legs) k (legs) (CKKW & Lönnblad, 2001) (Mangano, 2002) (+many more recent; see Alwall et al., EPJC53(2008)473) � 8 Peter Skands Monash University
THE GAIN THE COST Example: LHC 7 : W + 20-GeV Jets Example: e + e - → Z → Jets 2. Time to generate 1000 events W + N jets (Z → partons, fully showered & matched. No hadronization.) MLM w 3 rd order Matrix Elements 1000 SHOWERS Shower (w 1 st order MECs) 1000s SHERPA (CKKW-L) Time 100s 10s Matching Order 0 1 2 3 N JETS 1s 0.1s RATIO 2 3 4 5 6 Z → n : Number of Matched Emissions See e.g. Lopez-Villarejo & Skands, arXiv:1109.3608 Plot from mcplots.cern.ch; see arXiv:1306.3436 � 9 Peter Skands Monash University
3 : SUBTRACTION Examples: MC@NLO, aMC@NLO ๏ LO × Shower ๏ NLO X (2) X+1 (2) X (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) … X+1 (0) X+2 (0) X+3 (0) X+1 (0) X+2 (0) X+3 (0) Born … Born … … Fixed-Order Matrix Element … Shower Approximation 10 � Peter Skands Monash University
MATCHING 3: SUBTRACTION Examples: MC@NLO, aMC@NLO ๏ LO × Shower ๏ NLO - Shower NLO X (2) X+1 (2) X (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) … X+1 (0) X+2 (0) X+3 (0) X+1 (0) X+2 (0) X+3 (0) Born … Born … Expand shower approximation to … Fixed-Order Matrix Element NLO analytically, then subtract: Fixed-Order ME minus Shower … … Shower Approximation Approximation (NOTE: can be < 0!) 11 � Peter Skands Monash University
MATCHING 3: SUBTRACTION Examples: MC@NLO, aMC@NLO ๏ LO × Shower ๏ (NLO - Shower NLO ) × Shower X (2) X+1 (2) X (1) X (1) … … X+1 (1) X+2 (1) X+3 (1) X (1) … X (1) X (1) X (1) X (1) … X+1 (0) X+2 (0) X+3 (0) X+1 (0) X (1) X (1) Born … Born … Fixed-Order ME minus Shower … … Fixed-Order Matrix Element Approximation (NOTE: can be < 0!) Subleading corrections generated by … … Shower Approximation shower off subtracted ME 12 � Peter Skands Monash University
MATCHING 3: SUBTRACTION Examples: MC@NLO, aMC@NLO ๏ Combine ➤ MC@NLO Frixione, Webber, JHEP 0206 (2002) 029 • Consistent NLO + parton shower (though correction events can have w<0) • Recently, has been fully automated in aMC@NLO Frederix, Frixione, Hirschi, Maltoni, Pittau, Torrielli, JHEP 1202 (2012) 048 X (2) X+1 (2) … X+1 (1) X+2 (1) X+3 (1) X (1) … X+1 (0) X+2 (0) X+3 (0) Born … NB: w < 0 are a problem because they kill efficiency: Extreme example: 1000 positive-weight - 999 negative-weight events → statistical precision of 1 event, for 2000 generated (for comparison, normal MC@NLO has ~ 10% neg-weights) 13 � Peter Skands Monash University
POWHEG VS MC@NLO ๏ Both methods include the complete Example: Higgs Production 10 1 first-order (NLO) matrix elements. no damping no damping, LHEF • Difference is in whether only the h = m H / 1 . 2 GeV 10 0 h = m H / 2 GeV shower kernels are exponentiated h = 30 GeV (MC@NLO) or whether part of the T (pb/GeV) h = 30 GeV, LHEF 10 − 1 NLO matrix-element corrections are too No (POWHEG) 10 − 2 dp H Damping d σ Pure NLO ๏ In POWHEG, how much of the MEC 10 − 3 Plot from Bagnashi, Vicini, you exponentiate can be controlled JHEP 1601 (2016) 056 by the “hFact” parameter 10 − 4 0 50 100 150 200 250 300 350 400 p H T (GeV) • Variations basically span range h 2 between MC@NLO-like case, and D h = h 2 + ( p H ⊥ ) 2 original (hFact=1) POWHEG case (~ PYTHIA-style MECs) R s = D h R div R f = (1 � D h ) R div . , exponentiated not exponentiated 14 � Peter Skands Monash University
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