Towards Multijet Matching with Loops

Towards Multijet Matching with Loops
The Second Talk of the Workshop Towards Multijet Matching with Loops HP2.2, 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 … 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

ME-to-PS matching in VINCIA - 3
LL Shower Monte Carlos Arbitrary Process: X O: Observable {p} : momenta wX = |MX|2 or K|MX|2 S : Evolution operator Leading Order Pure Shower (all orders) 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 ME-to-PS matching in VINCIA - 3

“S” (for Shower) 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 “X + nothing” “X+something” A: splitting function ME-to-PS matching in VINCIA - 4

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

VINCIA Based on Dipole-Antennae So far:
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 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 Dipoles (=Antennae, not CS) – a dual description of QCD a Giele, Kosower, PS : PRD78(2008) Les Houches ‘NLM’ 2007 r b ME-to-PS matching in VINCIA - 6

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

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

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, ... 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

NLO with Addition First Order Shower expansion
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

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 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

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

Second Order Second Order Shower expansion for 4 partons (assuming first already matched) 1 2 3 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 min # of paths AR pT + AR recoil max # of paths DZ Everyone’s usual nightmare of a parton shower QE = pT QE = pT GGG AVG Vincia MAX Vincia MIN Vincia AVG ME-to-PS matching in VINCIA - 13

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

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

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

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

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

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

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

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

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

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

Ordering Number of paths in 4-parton phase space
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 min # of paths AR pT + AR recoil max # of paths DZ Mdaughter-dipole + AR recoil 3p-Thrust + 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

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

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 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

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

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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