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Designer Showers and Subtracted Matrix Elements

Freiburg, Apr 16 2008. Designer Showers and Subtracted Matrix Elements. Peter Skands CERN & Fermilab. Overview. Calculating collider observables Fixed order perturbation theory and beyond From inclusive to exclusive descriptions of the final state

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Designer Showers and Subtracted Matrix Elements

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  1. Freiburg, Apr 16 2008 Designer Showers and Subtracted Matrix Elements Peter Skands CERN & Fermilab

  2. Overview • Calculating collider observables • Fixed order perturbation theory and beyond • From inclusive to exclusive descriptions of the final state • Uncertainties and ambiguities beyond fixed order • The ingredients of a parton shower • A brief history of matching • New creations: Fall 2007 • A New Approach • Time-Like Showers Based on Dipole-Antennae • Some hopefully good news • VINCIA status and plans Time-Like Showers and Matching with Antennae - 2

  3. QuantumChromoDynamics • Main Tool: Matrix Elements calculated in fixed-order perturbative quantum field theory • Example: High-transverse momentum interaction Reality is more complicated Time-Like Showers and Matching with Antennae - 3

  4. Principal virtues Stochastic error O(N-1/2) independent of dimension Full (perturbative) quantum treatment at each order (KLN theorem: finite answer at each (complete) order) Monte Carlo at Fixed Order “Experimental” distribution of observable O in production of X: Fixed Order (all orders) {p} : momenta k : legs ℓ : loops “Monte Carlo”: N. Metropolis, first Monte Carlo calcultion on ENIAC (1948), basic idea goes back to Enrico Fermi High-dimensional problem (phase space) d≥5  Monte Carlo integration Note 1: For k larger than a few, need to be quite clever in phase space sampling Note 2: For ℓ > 0, need to be careful in arranging for real-virtual cancellations Time-Like Showers and Matching with Antennae - 4

  5. Parton Showers High-dimensional problem (phase space) d≥5  Monte Carlo integration + Formulation of fragmentation as a “Markov Chain”: A. A. Markov: Izvestiia Fiz.-Matem. Obsch. Kazan Univ., (2nd Ser.), 15(94):135 (1906) S:Evolution operator. Generates event, starting from {p}X • Hadronization: • iteration of X  X’ + hadron, according to phenomenological models (based on known properties of QCD, on lattice, and on fits to data). • Parton Showers: • iterative application of perturbatively calculable splitting kernels for n  n+1 partons Time-Like Showers and Matching with Antennae - 5

  6. Traditional Generators • Generator philosophy: • Improve Born-level perturbation theory, by including the ‘most significant’ corrections  complete events • Parton Showers • Hadronisation • The Underlying Event • Soft/Collinear Logarithms • Power Corrections • All of the above (+ more?) roughly (+ many other ingredients: resonance decays, beam remnants, Bose-Einstein, …) Asking for fully exclusive events is asking for quite a lot … Time-Like Showers and Matching with Antennae - 6

  7. Collider Energy Scales Hadron Decays Non-perturbative hadronisation, colour reconnections, beam remnants, non-perturbative fragmentation functions, pion/proton ratio, kaon/pion ratio, ... Soft Jets and Jet Structure Soft/collinear radiation (brems), underlying event (multiple perturbative 22 interactions + … ?), semi-hard brems jets, … Exclusive & Widths Resonance Masses… Hard Jet Tail High-pT jets at large angles Inclusive s • + Un-Physical Scales: • QF , QR : Factorization(s) & Renormalization(s) • QE : Evolution(s) Time-Like Showers and Matching with Antennae - 7

  8. Beyond Fixed Order e+e- 3 jets Problem 1: bremsstrahlung corrections are singular for soft/collinear configurations  spoils fixed-order truncation Time-Like Showers and Matching with Antennae - 8

  9. Diagrammatical Explanation 1 dσX+2 “DLA” α sab saisib • dσX = … • dσX+1 ~ dσX g2 2 sab /(sa1s1b) dsa1ds1b • dσX+2 ~ dσX+1 g2 2 sab/(sa2s2b) dsa2ds2b • dσX+3 ~ dσX+2 g2 2 sab/(sa3s3b) dsa3ds3b dσX dσX+1 dσX+2 This is an approximation of inifinite-order tree-level cross sections • But it’s not yet an “evolution” • What’s the total cross section we would calculate from this? • σX;tot = int(dσX) + int(dσX+1) + int(dσX+2) + ... Probability not conserved, events “multiply” with nasty singularities! Just an approximation of a sum of trees. But wait, what happened to the virtual corrections? KLN? Time-Like Showers and Matching with Antennae - 9

  10. Diagrammatical Explanation 2 dσX+2 “DLA” α sab saisib • dσX = … • dσX+1 ~ dσX g2 2 sab /(sa1s1b) dsa1ds1b • dσX+2 ~ dσX+1 g2 2 sab/(sa2s2b) dsa2ds2b • dσX+3 ~ dσX+2 g22 sab/(sa3s3b) dsa3ds3b +Unitarisation:σtot = int(dσX)  σX;PS= σX - σX+1 - σX+2- … dσX dσX+1 dσX+2 Given a jet definition, an event has either 0, 1, 2, or … jets • Interpretation: the structure evolves! (example: X = 2-jets) • Take a jet algorithm, with resolution measure “Q”, apply it to your events • At a very crude resolution, you find that everything is 2-jets • At finer resolutions  some 2-jets migrate  3-jets =σX+1(Q) = σX;incl– σX;excl(Q) • Later, some 3-jets migrate further, etc  σX+n(Q) = σX;incl– ∑σX+m<n;excl(Q) • This evolution takes place between two scales, Qin and Qfin = QF;ME and Qhad • σX;PS = int(dσX) - int(dσX+1) - int(dσX+2) + ... = int(dσX) EXP[ - int(α 2sab /(sa1s1b) dsa1 ds1b ) ] Time-Like Showers and Matching with Antennae - 10

  11. Beyond Fixed Order Fixed Order (all orders) • Evolution Operator, S (as a function of “time” t=1/Q) • “Evolves” phase space point: X  … • Can include entire (interleaved) evolution, here focus on showers • Observable is evaluated on final configuration • S unitary (as long as you never throw away an event) •  normalization of total (inclusive)σ unchanged (σLO,σNLO, σNNLO, σexp, …) • Only shapes are predicted (i.e., also σ after shape-dependent cuts) wX : |MX|2 S : Evolution operator {p} : momenta Pure Shower (all orders) Time-Like Showers and Matching with Antennae - 11

  12. Perturbative Evolution “X + nothing” “X+something” wX : |MX|2 S : Evolution operator {p} : momenta Pure Shower (all orders) • Evolution Operator, S (as a function of “time” t=1/Q) • Defined in terms of Δ(t1,t2) – The integrated probability the system does not change state between t1 and t2(Sudakov) A: splitting function Analogous to nuclear decay: Time-Like Showers and Matching with Antennae - 12

  13. Constructing LL Showers • The final answer will depend on: • The choice of evolution variable • The splitting functions (finite terms not fixed) • The phase space map ( dΦn+1/dΦn ) • The renormalization scheme (argument of αs) • The infrared cutoff contour (hadronization cutoff) • They are all “unphysical”, in the same sense as QFactorizaton, etc. • At strict LL, any choice is equally good • However, 20 years of parton showers have taught us: many NLL effects can be (approximately) absorbed by judicious choices • Effectively, precision is much better than strict LL, but still not formally NLL • E.g., (E,p) cons., “angular ordering”, using pT as scale in αs, with ΛMSΛMC, …  Clever choices good for process-independent things, but what about the process-dependent bits? … + matching Time-Like Showers and Matching with Antennae - 13

  14. Matching • Traditional Approach: take the showers you have, expand them to 1st order, and fix them up • Sjöstrand (1987): Introducere-weightingfactor on first emission  1st order tree-level matrix element (ME) (+ further showering) • Seymour (1995): + where shower is “dead”, add separate events from 1st order tree-level ME, re-weighted by “Sudakov-like factor” (+ further showering) • Frixione & Webber (2002):Subtract1st order expansion from 1st order tree and 1-loop ME  add remainder ME correction events (+ further showering) • Multi-leg Approaches (Tree level only): • Catani, Krauss, Kuhn, Webber (2001): Substantial generalization of Seymour’s approach, to multiple emissions, slicingphase space into “hard”  M.E. ; “soft”  P.S. • Mangano (?): pragmatic approach to slicing: after showering, match jets to partons, reject events that look “double counted” A brief history of conceptual breakthroughs in widespread use today: Time-Like Showers and Matching with Antennae - 14

  15. New Creations: Fall 2007 • Showers designed specifically for matching • Nagy, Soper (2006):Catani-Seymour showers • Dinsdale, Ternick, Weinzierl (Sep 2007) & Schumann, Krauss (Sep 2007): implementations • Giele, Kosower, PS (Jul 2007): Antenna showers • (incl. implementations) • Other new showers: partially designed for matching • Sjöstrand (Oct 2007): New interleaved evolution of FSR/ISR/UE • Official release of Pythia8 last week • Webber et al (HERWIG++): Improved angular ordered showers • Winter, Krauss (Dec 2007) : Dipole-antenna showers • (incl. implementation in SHERPA.) Similar to ARIADNE, but more antenna-like for ISR • Nagy, Soper (Jun 2007 + Jan 2008):Quantum showers •  subleading color, polarization (so far no implementation) • New matching proposals • Nason (2004): Positive-weight variant of MC@NLO • Frixione, Nason, Oleari (Sep 2007): Implementation: POWHEG • Giele, Kosower, PS (Jul 2007):Antenna subtraction • VINCIA + an extension of that I will present here for the first time Time-Like Showers and Matching with Antennae - 15

  16. Some Holy Grails • Matching to first order + (N)LL ~ done • 1st order : MC@NLO, POWHEG, PYTHIA, HERWIG • Multi-leg tree-level: CKKW, MLM, … (but still large uncertainties) • Simultaneous 1-loop and multi-leg matching • 1st order : NLO (Born) + LO (Born + m) + (N)LL (Born + ∞) • 2nd order : NLO (Born+1) + LO (Born + m) + (N)LL (Born + ∞) • Showers that systematically resum higher logs • (N)LL  NLL  NNLL  … ? • (N)LC  NLC  … ? • Solving any of these would be highly desirable • Solve all of them ? • NNLO (Born) + LO (Born + m) + (N)NLL + string-fragmentation • + reliable uncertainty bands Time-Like Showers and Matching with Antennae - 16

  17. Parton Showers • The final answer depends on: • The choice of evolution variable • The splitting functions (finite/subleading terms not fixed) • The phase space map ( dΦn+1/dΦn ) • The renormalization scheme (argument of αs) • The infrared cutoff contour (hadronization cutoff) • Step 1, Quantify uncertainty: vary all of these (within reasonable limits) • Step 2, Systematically improve: Understand the importance of each and how it is canceled by • Matching to fixed order matrix elements, at LO, NLO, NNLO, … • Higher logarithms, subleading color, etc, are included • Step 3, Write a generator: Make the above explicit (while still tractable) in a Markov Chain context  matched parton shower MC algorithm Time-Like Showers and Matching with Antennae - 17

  18. Based on Dipole-Antennae Shower off color-connected pairs of partons Plug-in to PYTHIA 8.1 (C++) So far: 3 different shower evolution variables: pT-ordering (= ARIADNE ~ PYTHIA 8) Dipole-mass-ordering (~ but not = PYTHIA 6, SHERPA) Thrust-ordering (3-parton Thrust) For each: an infinite family of antenna functions Laurent series in branching invariants with arbitrary finite terms Shower cutoff contour: independent of evolution variable IR factorization “universal” Several different choices for αs (evolution scale, pT, mother antenna mass, 2-loop, …) Phase space mappings: 2 different choices implemented Antenna-like (ARIADNE angle) or Parton-shower-like: Emitter + longitudinal Recoiler VINCIA VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE Gustafson, PLB175(1986)453; Lönnblad (ARIADNE), CPC71(1992)15. Azimov, Dokshitzer, Khoze, Troyan, PLB165B(1985)147 Kosower PRD57(1998)5410; Campbell,Cullen,Glover EPJC9(1999)245 Dipoles (=Antennae, not CS) – a dual description of QCD a Giele, Kosower, PS : hep-ph/0707.3652 + Les Houches 2007 r b Time-Like Showers and Matching with Antennae - 18

  19. Dipole-Antenna Showers • Dipole branching and phase space ( Most of this talk, including matching by antenna subtraction, should be applicable to ARIADNE and the SHERPA dipole-shower as well) Giele, Kosower, PS : hep-ph/0707.3652 Time-Like Showers and Matching with Antennae - 19

  20. Dipole-Antenna Functions • Starting point: “GGG” antenna functions, e.g., ggggg: • Generalize to arbitrary double Laurent series:  Can make shower systematically “softer” or “harder” • Will see later how this variation is explicitly canceled by matching •  quantification of uncertainty •  quantification of improvement by matching Gehrmann-De Ridder, Gehrmann, Glover, JHEP 09 (2005) 056 yar = sar / si si = invariant mass of i’th dipole-antenna Frederix, Giele, Kosower, PS : Les Houches NLM, arxiv:0803.0494 Singular parts fixed, finite terms arbitrary Time-Like Showers and Matching with Antennae - 20

  21. Comparison Frederix, Giele, Kosower, PS : Les Houches ‘NLM’, arxiv:0803.0494 Time-Like Showers and Matching with Antennae - 21

  22. Quantifying Matching • The unknown finite terms are a major source of uncertainty • DGLAP has some, GGG have others, ARIADNE has yet others, etc… • They are arbitrary (and in general process-dependent  don’t tune!) Varying finite terms only with αs(MZ)=0.137, μPS=pT, pThad = 0.5 GeV (huge variation with μPS from pureLL point of view, but NLL tells you using pT at LL  (N)LL. Formalize that.) Time-Like Showers and Matching with Antennae - 22

  23. Tree-level matching to X+1 • Expand parton shower to 1st order (real radiation term) • Matrix Element (Tree-level X+1 ; above thad)  Matching Term (= correction events to be added) •  variations in finite terms (or dead regions) in Aicanceled (at this order) • (If A too hard, correction can become negative  negative weights) Inverse phase space map ~ clustering Giele, Kosower, PS : hep-ph/0707.3652 Time-Like Showers and Matching with Antennae - 23

  24. Matching by Reweighted Showers wX : |MX|2 S : Evolution operator {p} : momenta • Go back to original shower definition • Possible to modify S to expand to the “correct” matrix elements ? Pure Shower (all orders) 1st order: yes Generate an over-estimating (trial) branching Reweight it by vetoing it with the probability Sjöstrand, Bengtsson : Nucl.Phys.B289(1987)810; Phys.Lett.B185(1987)435 Norrbin, Sjöstrand : Nucl.Phys.B603(2001)297 w>0 as long as |M|2 > 0 But2nd and beyond difficult due to lack of clean PS expansion Time-Like Showers and Matching with Antennae - 24

  25. Towards NNLO + NLL • Basic idea: extend reweigthing to 2nd order • 23 tree-level antennae  NLO • 23 one-loop + 24 tree-level antennae  NNLO • And exponentiate it • Exponentiating 23 (dipole-antenna showers)  (N)LL • Complete NNLO captures the singularity structure up to (N)NLL • So a shower incorporating all these pieces exactly should be able to reach NLL resummation, with a good approximation to NNLL; + exact matching up to NNLO should be possible • Start small, do it for leading-color first, included the qqbar 24 antennae, A04 , B04 . • Gives exact matching of Z4 since these happen to be the exact matrix elements for that process. • Still missing the remaining 24 functions, matching to the running coupling in one-loop 23, and inclusion of next-to-leading color • Full one-loop 23 matching (i.e., the finite terms for Z decay) Time-Like Showers and Matching with Antennae - 25

  26. 24 Matching by reweighting • Starting point: • LL shower w/ large coupling and large finite terms to generate “trial” branchings (“sufficiently” large to over-estimate the full ME). • Accept branching [i] with a probability • Each point in 4-parton phase space then receives a contribution • Also need to take into account ordering cancellation of dependence 1st order matching term (à la Sjöstrand-Bengtsson) 2nd order matching term (with 1st order subtracted) (If you think this looks deceptively easy, you are right) Time-Like Showers and Matching with Antennae - 26

  27. 23 one-loop Matching by reweighting • Unitarity of the shower  effective 2nd order 3-parton term contains • An integral over A04 over the 34 phase space below the 3-parton evolution scale (all the way from QE3 to 0) • An integral over the 23 antenna function above the 3-parton evolution scale (from MZ to QE3) • (These two combine to give the an evolution-dependence, canceled by matching to the actual 3-parton 1-loop ME) • A term coming from the expansion of the 23 αs(μPS) • Combine with 34 evolution to cancel scale dependence • A term coming from a tree-level branching off the one-loop 2-parton correction. • It then becomes a matter of collecting these pieces and subtracting them off, e.g., A13 . • After cancellation of divergences, an integral over the shower-subtracted A04 remains  Numerical? No need to exponentiate  must be evaluated once per event. The other pieces (except αs) are already in the code. Time-Like Showers and Matching with Antennae - 27

  28. Tree-level 23 + 24 in Action • The unknown finite terms are a major source of uncertainty • DGLAP has some, GGG have others, ARIADNE has yet others, etc… • They are arbitrary (and in general process-dependent) Varying finite terms only with αs(MZ)=0.137, μR=pT, pThad = 0.5 GeV Time-Like Showers and Matching with Antennae - 28

  29. LEP Comparisons Still with αs(MZ)=0.137 : THE big thing remaining … Time-Like Showers and Matching with Antennae - 29

  30. What to do next? • Further shower studies • Include the remaining 4-parton antenna functions • Measuring, rather than tuning, hadronization? • Go further with one-loop matching • Include exact running coupling from 3-parton one-loop • + Exponentiate • Include full 3-parton one-loop (i.e., including finite terms) •  Shower Monte Carlo at NNLO + NLL • Extend to the initial state • The Krauss-Winter shower looks close; we would concentrate on the uncertainties and matching. • Extend to massive particles • Massive antenna functions, phase space, and evolution (+matching?) Time-Like Showers and Matching with Antennae - 30

  31. Extra Material Frederix, Giele, Kosower, PS : Les Houches Proc., in preparation • Number of partons and number of quarks • Nq shows interesting dependence on ordering variable Time-Like Showers and Matching with Antennae - 31

  32. The Bottom Line HQET FO DGLAP • The S matrix is expressible as a series in gi, gin/Qm, gin/xm, gin/mm, gin/fπm, … • To do precision physics: • Solve more of QCD • Combine approximations which work in different regions: matching • Control it • Good to have comprehensive understanding of uncertainties • Even better to have a way to systematically improve • Non-perturbative effects • don’t care whether we know how to calculate them BFKL χPT Time-Like Showers and Matching with Antennae - 32

  33. Matching “X + nothing” “X+something” wX : |MX|2 S : Evolution operator {p} : momenta Pure Shower (all orders) A: splitting function Matched shower (including simultaneous tree- and 1-loop matching for any number of legs) Loop-level “virtual+unresolved” matching term for X+k Tree-level “real” matching term for X+k Giele, Kosower, PS : hep-ph/0707.3652 Time-Like Showers and Matching with Antennae - 33

  34. Example: Z decays • VINCIA and PYTHIA8 (using identical settings to the max extent possible) αs(pT), pThad = 0.5 GeV αs(mZ) = 0.137 Nf = 2 Note: the default Vincia antenna functions reproduce the Z3 parton matrix element; Pythia8 includes matching to Z3 Frederix, Giele, Kosower, PS : Les Houches NLM, arxiv:0803.0494 Time-Like Showers and Matching with Antennae - 34

  35. Example: Z decays • Why is the dependence on the evolution variable so small? • Conventional wisdom: evolution variable has huge effect • Cf. coherent vs non-coherent parton showers, mass-ordered vs pT-ordered, etc. • Dipole-Antenna showers resum radiation off pairs of partons •  interference between 2 partons included in radiation function • If radiation function = dipole formula  intrinsically coherent • Remaining dependence on evolution variable much milder than for conventional showers • The main uncertainty in this case lies in the choice of radiation function away from the collinear and soft regions •  dipole-antenna showers under the hood … Gustafson, PLB175(1986)453 Time-Like Showers and Matching with Antennae - 35

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