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Towards Multijet Matching with Loops

The Second Talk of the Workshop. Towards Multijet Matching with Loops. HP 2 .2, Buenos Aires, October 2007. Precision Chromodynamics. Monte Carlo problem Uncertainty on fixed orders and logs obscures clear view on hadronization and the underlying event So we just need …

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Towards Multijet Matching with Loops

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  1. The Second Talk of the Workshop Towards Multijet Matching with Loops HP2.2, Buenos Aires, October 2007

  2. Precision Chromodynamics • Monte Carlo problem • Uncertainty on fixed orders and logs obscures clear view on hadronization and the underlying event • So we just need … • An NNLO + NLO multileg + NLL Monte Carlo, with uncertainty bands, please • Then … • We could see hadronization and UE clearly  solid models   Energy Frontier Intensity Frontier The Astro Guys Precision Frontier Anno 2018 The Tevatron and LHC data will be all the energy frontier data we’ll have for a long while ME-to-PS matching in VINCIA - 2

  3. Evolution Operator, S “Evolves” phase space point: X  … As a function of “time” t=1/Q Observable is evaluated on final configuration S unitary (as long as you never throw away or reweight an event)  normalization of total (inclusive)σ unchanged (σLO,σNLO, σNNLO, σexp, …) Only shapes are predicted (i.e., also σ after shape-dependent cuts) Can expand S to any fixed order (for given observable) Can check agreement with ME Can do something about it if agreement less than perfect: reweight or add/subtract Arbitrary Process: X LL Shower Monte Carlos O: Observable {p} : momenta wX = |MX|2 or K|MX|2 S : Evolution operator Leading Order Pure Shower (all orders) ME-to-PS matching in VINCIA - 3

  4. “S” (for Shower) “X + nothing” “X+something” • Evolution Operator, S (as a function of “time” t=1/Q) • Defined in terms of Δ(t1,t2)(Sudakov) • The integrated probability the system does not change state between t1 and t2 • NB: Will not focus on where Δ comes from here, just on how it expands • = Generating function for parton shower Markov Chain A: splitting function ME-to-PS matching in VINCIA - 4

  5. Constructing LL Showers • The final answer will depend on: • The choice of evolution “time” • The splitting functions (finite terms not fixed) • The phase space map (“recoils”, dΦn+1/dΦn ) • The renormalization scheme (argument of αs) • The infrared cutoff contour (hadronization cutoff) Variations  Comprehensive uncertainty estimates (showers with uncertainty bands) ME-to-PS matching in VINCIA - 5

  6. Based on Dipole-Antennae Shower off color-connected pairs of partons Plug-in to PYTHIA 8 (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, …) 3 different phase space maps Ariadne or Kosower “antenna” recoils, or 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 : PRD78(2008)014026 + Les Houches ‘NLM’ 2007 r b ME-to-PS matching in VINCIA - 6

  7. Example: Jet Rates • The unknown finite terms are important • They are arbitrary (and in general process-dependent) • Uncertainty in hard region already at first order • Cascade down to produce uncontrolled tower of subleading logs Varying finite terms only with αs(MZ)=0.137, μR=pT, pThad = 0.5 GeV ME-to-PS matching in VINCIA - 7

  8. Constructing LL Showers • The final answer will depend on: • The choice of evolution “time” • The splitting functions (finite terms not fixed) • The phase space map (“recoils”, 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 • We’ve learned, however: some NLL effects can be (approximately) absorbed by judicious choices • E.g., (E,p) cons., coherence, using pT as scale in αs, with ΛMSΛMC, … • Effectively, precision is better than strict LL, but still not formally NLL Variations  Comprehensive uncertainty estimates (showers with uncertainty bands) •  Clever choices fine (for process-independent things), can we do better? … + matching ME-to-PS matching in VINCIA - 8

  9. Matching in a nutshell • There are two fundamental approaches • Additive • Multiplicative • Most current approaches based on addition, in one form or another • Herwig (Seymour, 1995), but also CKKW, MLM, MC@NLO, ... • In these approaches, you add event samples with different multiplicities • Need separate ME samples for each multiplicity. Relative weights a priori unknown. • The job is to construct weights for them, and modify/veto the showers off them, to avoid double counting of both logs and finite terms • But you can also do it by multiplication • Pythia (Sjöstrand, 1987): modify only the shower • All events start as Born + reweight at each step. • Using the shower as a weighted phase space generator •  only works for showers with NO DEAD ZONES • The job is to construct reweighting coefficients • Complicated shower expansions  only first order so far • Generalized to include 1-loop first-order  POWHEG Seymour, Comput.Phys.Commun.90(1995)95 Sjöstrand, Bengtsson : Nucl.Phys.B289(1987)810; Phys.Lett.B185(1987)435 Norrbin, Sjöstrand : Nucl.Phys.B603(2001)297 Massive Quarks All combinations of colors and Lorentz structures ME-to-PS matching in VINCIA - 9

  10. NLO with Addition Multiplication at this order  A = |M3|2/|M2|2 • First Order Shower expansion PS Unitarity of shower  3-parton real = ÷ 2-parton “virtual” • 3-parton real correction (GGG + example finite terms; α, β) Finite terms cancel in 3-parton O • 2-parton virtual correction (same example) Finite terms cancel in 2-parton O (normalization) ME-to-PS matching in VINCIA - 10

  11. Matching to X+1: Tree-level • Herwig • In dead zone: Ai = 0 add events corresponding to unsubtracted |MX+1| • Outside dead zone: reweighted à la Pythia  Ai = |MX+1| •  no additive correction necessary • CKKW and L-CKKW • At this order identical to Herwig, with “dead zone” for kT > kTcut introduced by hand • MC@NLO • In dead zone: identical to Herwig • Outside dead zone: AHerwig >|MX+1| wX+1 negative  negative weights • Pythia • Ai = |MX+1| over all of phase space  no additive correction necessary • Powheg • At this order identical to Pythia •  no negative weights HERWIG TYPE PYTHIA TYPE ME-to-PS matching in VINCIA - 11

  12. Matching in Vincia • We are pursuing three strategies in parallel • Addition (aka subtraction) • Simplest, but has generic negative weights and hard to exponentiate corrections • Guaranteed to fill all of phase space (unsubtracted ME in dead regions) • Multiplication (aka reweighting) • Complicated, so 1-loop matching difficult beyond first order, but has generic positive weights and “automatically” exponentiates  path to NLL • Only fills phase space populated by shower: dead zones problematic • Hybrid • Trying to combine simple expansions with positive weights, full phase space, and exponentiation • Goal • Multi-leg “plug-and-play” NLO + “improved”-LL shower Monte Carlo • Including uncertainty bands (exploring uncontrolled terms) • Extension to NNLO + NLL ? ME-to-PS matching in VINCIA - 12

  13. Second Order 0 1 2 3 AR pT + AR recoil max # of paths DZ min # of paths • Second Order Shower expansion for 4 partons (assuming first already matched) • Problem 1: dependence on evolution variable • Shower is ordered  t4 integration only up to t3 •  2, 1, or 0 allowed “paths” • Dead zone not good for multiplication QE = pT(i,j,k) = mijmjk/mijk Everyone’s usual nightmare of a parton shower QE = pT QE = pT Vincia MAX GGG AVG Vincia MIN Vincia AVG ME-to-PS matching in VINCIA - 13

  14. Second Order with Unordered Showers • For multiplication: allow power-suppressed “unordered” branchings GGG Uord AVG Vincia Uord MAX Vincia Uord AVG Vincia Uord MIN • Removes dead zone + better approx than fully unordered • (Good initial guess  better reweighting efficiency) • Problem 2: leftover Subleading Logs • There are still unsubtractred subleading divergences in the ME ME-to-PS matching in VINCIA - 14

  15. Leftover Logs • Most obvious for subtraction in Dead Zone • ME completely unsubtracted in Dead Zone  leftovers • But also true in general: the shower is still formally LL everywhere • NLL leftovers are unavoidable • Additional sources: Subleading color, Polarization • Beat them or join them? • Beat them: not resummed •  brute force regulate with Theta (or smooth) function ~ CKKW “matching scale” • Join them: absorb leftovers systematically in shower resummation • But looks like we would need polarized NLL-NLC showers … ! • Could take some time … • In the meantime, maybe we can cheat … (don’t stop matching)! Note: more legs  more logs, so ultimately will still need regulator. But try to postpone to NNLL level. ME-to-PS matching in VINCIA - 15

  16. 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 Sjöstrand-Bengtsson term 2nd order matching term (with 1st order subtracted out) (If you think this looks deceptively easy, you are right) Note: to maintain positivity for subleading colour, need to match across 4 events, 2 representing one color ordering, and 2 for the other ordering ME-to-PS matching in VINCIA - 16

  17. General 2nd Order (& NLL Matching) • Include unitary shower (S) and non-unitary “K-factor” (K) corrections • S: branching probability modification, goes back into Sudakov  resummed • All logs should be here. • Unitary  does not modify normalization • The simpler the better : will explicitly appear in 1-loop subtractions • The simpler the better : will need to be evaluated once for every nested 24 branching (if NLL) • K: event weight modification, does not go back into Sudakov  not resummed • Finite corrections can go here ( + regulated logs) • Non-unitary  changes normalization (“K” factors) • Can be arbitrarily complicated: will not appear in 1-loop subtractions (?) • Can be arbitrarily complicated: will only need to be evaluated once per event • With this notation, • Addition/Subtraction: S = 1, K ≠ 1 • Multiplication/Reweighting: K = 1, S≠ 1 • Hybrid: S contains logs (kept as simple as possible), K contains the rest (stick complicated stuff here) ME-to-PS matching in VINCIA - 17

  18. The Z3 1-loop term • Second order matching term for 3 partons • Additive (S=1)  Ordinary NLO subtraction + shower leftovers • Shower off w2(V) • “Coherence” term: difference between 2- and 3-parton (power-suppressed) evolution above QE3. Explicit QE-dependence cancellation. • δα: Difference between alpha used in shower (μ = pT) and alpha used for matching  Explicit scale choice cancellation • Integral over w4(R) in IR region still contains NLL divergences  regulate • Logs not resummed, so remaining (NLL) logs in w3(R)also need to be regulated • Multiplicative : S = (1+…)  Modified NLO subtraction + shower leftovers • A*S contains all logs from tree-level  w4(R) finite. • Any remaining logs in w3(V) cancel against NNLO  NLL resummation if put back in S ME-to-PS matching in VINCIA - 18

  19. VINCIA in Action: Jet Rates • The unknown finite terms are important • They are arbitrary (and in general process-dependent) • Uncertainty in hard region already at first order • Cascade down to produce uncontrolled tower of subleading logs Varying finite terms only with αs(MZ)=0.137, μR=pT, pThad = 0.5 GeV ME-to-PS matching in VINCIA - 19

  20. VINCIA in Action: LEP Still with αs(MZ)=0.137 : THE big thing remaining … ME-to-PS matching in VINCIA - 20

  21. VINCIA in Action: LEP Still with αs(MZ)=0.137 : THE big thing remaining … ME-to-PS matching in VINCIA - 21

  22. VINCIA in Action: LEP Still with αs(MZ)=0.137 : THE big thing remaining … ME-to-PS matching in VINCIA - 22

  23. The next big steps • Z3 at one loop • Opens multi-parton matching at 1 loop • Required piece in NNLO Z matching • Allows to get a fix on Sudakov terms generated by unordering • Allows to get a fix on running coupling • Work in progress • Write up complete framework for additive matching •  NLO Z3 and NNLO matching within reach • Derivations not yet finished for multiplicative matching … • Complete NLL showers slightly further down the road • Turn to the initial state, massive particles, other NLL effects ME-to-PS matching in VINCIA - 23

  24. Overview • LL Shower Monte Carlos • Constructing LL Showers: Uncertainties at LL • The VINCIA Antenna Showers • Matching • Multileg Matching 1: Additive (subtraction) • Simple subtraction terms • Positive and Negative weights • Subleading Logs not resummed  need explicit regulators • Multileg Matching 2: Multiplicative (reweighting) • Positive weights • Phase space coverage  unordered showers (power-suppressed) • Exponentiated matching to 24: towards NLL showers • Complicated subtraction terms • Multileg Matching 3: Hybrid (subtraction + some reweighting) • Best of both? • Towards NNLO matching and beyond ME-to-PS matching in VINCIA - 24

  25. Differences • Addition • Weight(X+n) = ME(x+n) – Shower(X,X+1,X+2,…,X+n-1) • Weight can have either sign  negative weights (even at tree level) • Special case 1: dead zones  weight = ME • Necessary in HERWIG  Seymour’s 1995 paper • Utilized in CKKW etc: force dead zones  simpler matching, no negative weights • Special case 2: shower function = ME(x+n)/ME(x+n-1) • POWHEG: ensures Weight(X+n) = 0 and Weight(X+n-1) ~ KNLO * MELO • Multiplication • Reweight(X+n) = ME(X+n) / Shower(X+n-1) • Physical matrix elements positive  Reweight > 0 • Shower evolution is unitary • Sudakov contains ME (as in Pythia, Powheg) •  complicated subtractions beyond first order ME-to-PS matching in VINCIA - 25

  26. Ordering AR pT + AR recoil max # of paths DZ min # of paths 0 • Number of paths in 4-parton phase space • Starting at 2-parton scale = 100 GeV • X- and Y-axes = pT(0,1,2) and pT(1,2,3) • So each (X,Y) bin contains many 4-parton PS points • 10M 4-parton points generated with Rambo: test ordering 1 pT(i,j,k) = mijmjk/mijk 2 3 3p-Thrust + AR recoil Mdaughter-dipole + AR recoil AR pT + “longitudinal” recoil 3p-Thrust + “longitudinal” recoil Q2-ordering + AR recoil pT = mijmjk/mijk Mdd = min(mij,mjk) M(1-T3) = min(mij,mjk,mik) Q = max(mij,mjk) D.Z. D.Z. ME-to-PS matching in VINCIA - 26

  27. Ratio: Showers / Z4 ME AVG MIN/MAX Alternative QE… (GGG/Vincia difference: Vincia only includes nestings of(23)that are ordered in the shower evolution variable) QE = pT QE = pT QE = T3 Everyone’s usual nightmare of a parton shower Vincia MAX GGG AVG Vincia MIN Vincia AVG QE = pT C=0 QE = pT C=0 QE = T3 C=0 Vincia MAX GGG no C AVG Finite Terms = 0 Vincia MIN Vincia AVG ME-to-PS matching in VINCIA - 27

  28. Why NLO “multileg”? • Including X at one loop  “NLO” matching ? = NLO only for distributions that are not a δ at LO (e.g., yX) = LO for any distribution that “starts” at X+1 (e.g., pTX) = “Improved” LL for any distribution that “starts” at X+2 (e.g., 2-jet rates) • Perturbative series still barely under control • Combining MC@NLO with CKKW  NLO + multi-leg tree ? = NLO only for distributions that are not a δ at LO = LO for any distribution that “starts” at X+1, … X+N = “improved”LL for any distribution that “starts” beyond X+N • NLO N-jet precision can only be accessed by NLO multileg ME-to-PS matching in VINCIA - 28

  29. 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 ME-to-PS matching in VINCIA - 29

  30. 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, if ordered, or from sij to 0 if unordered ) • 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. ME-to-PS matching in VINCIA - 30

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