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  1. Eugene, February 2009 Higher Order Aspects of Parton Showers

  2. 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 calculation 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 k+ℓ > 0, need to be careful in arranging for real-virtual cancellations A Guide to Hadron Collisions - 2

  3. Bremsstrahlung Example: SUSY @ LHC LHC - sps1a - m~600 GeV Plehn, Rainwater, PS PLB645(2007)217 FIXED ORDER pQCD inclusiveX + 1 “jet” inclusiveX + 2 “jets” Cross section for 1 or more 50-GeV jets larger than total σ, obviously non-sensical (Computed with SUSY-MadGraph) • Naively, brems suppressed byαs ~ 0.1 • Truncate at fixed order = LO, NLO, … • However, if ME >> 1 can’t truncate! • Example: SUSY pair production at 14 TeV, with MSUSY ~ 600 GeV • Conclusion: 100 GeV can be “soft” at the LHC • Matrix Element (fixed order) expansion breaks completely down at 50 GeV • With decay jets of order 50 GeV, this is important to understand and control A Guide to Hadron Collisions - 3

  4. Beyond Fixed Order 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 a parton shower, 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) + ... Just an approximation of a sum of trees  no real-virtual cancellations But wait, what happened to the virtual corrections? KLN? KLN guarantees that sing{int(real)} = ÷ sing{virtual} approximate virtual = int(real) A Guide to Hadron Collisions - 4

  5. Beyond Fixed Order 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;excl= 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 ~ s and Qend = Qhad • σX;excl = int(dσX) - int(dσX+1,2,3,…;excl) = int(dσX) EXP[ - int(dσX+1 / dσX) ] • σX;tot = Sum (σX+0,1,2,3,…;excl ) = int(dσX) A Guide to Hadron Collisions - 5

  6. 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) A Guide to Hadron Collisions - 6

  7. “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 A Guide to Hadron Collisions - 7

  8. Constructing LL Showers • In the previous slide, you saw many dependencies on things not traditionally found in matrix-element calculations: • 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 (vertex-by-vertex argument of αs) • The infrared cutoff contour (hadronization cutoff) Variations  Comprehensive uncertainty estimates (showers with uncertainty bands) Matching to MEs (& NnLL?) Reduced Dependence (systematic reduction of uncertainty) A Guide to Hadron Collisions - 8

  9. A (complete idiot’s) Solution? X inclusive X exclusive ≠ X+1 inclusive X+1 exclusive X+2 inclusive X+2 inclusive • Combine different starting multiplicites •  inclusive sample? • In practice – Combine • [X]ME+ showering • [X + 1 jet]ME+ showering • … • Doesn’t work • [X] + shower is inclusive • [X+1] + shower is also inclusive Run generator for X (+ shower) Run generator for X+1 (+ shower) Run generator for … (+ shower) Combine everything into one sample What you want What you get Overlapping “bins” One sample A Guide to Hadron Collisions - 9

  10. The Matching Problem • [X]ME+ showeralready containssing{[X + n jet]ME} • So we really just missed the non-LL bits, not the entire ME! • Adding full [X + n jet]MEis overkill LL singular terms are double-counted • Solution 1: work out the difference and correct by that amount •  add “shower-subtracted” matrix elements • Correction events with weights : wn = [X + n jet]ME – Shower{wn-1,2,3,..} • I call these matching approaches “additive” • Solution 2: work out the ratio between PS and ME •  multiply shower kernels by that ratio (< 1 if shower is an overestimate) • Correction factor on n’th emission Pn = [X + n jet]ME / Shower{[X+n-1 jet]ME} • I call these matching approaches “multiplicative” A Guide to Hadron Collisions - 10

  11. Matching in a nutshell • There are two fundamental approaches • Additive • Multiplicative • Most current approaches based onaddition, in one form or another • Herwig(Seymour, 1995), but also CKKW, MLM, MC@NLO, ... • 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 bymultiplication • 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 A Guide to Hadron Collisions - 11

  12. 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 A Guide to Hadron Collisions - 12

  13. Based on Dipole-Antennae Shower off color-connected pairs of partons Plug-in to PYTHIA 8 (C++) So far: Choice of evolution time: pT-ordering Dipole-mass-ordering Thrust-ordering Splitting functions QCD singular terms + arbitrary finite terms (Taylor series) Phase space map Antenna-like or Parton-shower-like Renormalization scheme (μR = {evolution scale, pT, s, 2-loop, …} ) Infrared cutoff contour (hadronization cutoff) Same options as for evolution time, but independent of time  universal choice 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 A Guide to Hadron Collisions - 13

  14. Ordering Phase Space for 23 kT m2 collinear Partitioned-Dipole Eg Angle soft pT (Ariadne) mant 1-T Dipole-Antenna A Guide to Hadron Collisions - 14

  15. 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” • 0 = Dead Zone : not good for reweighting QE = pT(i,j,k) = mijmjk/mijk A Guide to Hadron Collisions - 15

  16. Second OrderAVERAGEs of Over/Under-counting 0 1 2 3 • Second Order Shower expansion for 4 partons (assuming first already matched) Define over/under-counting ratio: PStree / MEtree NB: AVERAGE of R4 distribution A Guide to Hadron Collisions - 16

  17. Second OrderEXTREMA of Over/Under-counting 0 1 2 3 • Second Order Shower expansion for 4 partons (assuming first already matched) Define over/under-counting ratio: PStree / MEtree NB: EXTREMA of R4 distribution (100M points) A Guide to Hadron Collisions - 17

  18. (Stupid Choices) A Guide to Hadron Collisions - 18

  19. Dependence on Finite Terms • Antenna/Dipole/Splitting functions are ambiguous by finite terms A Guide to Hadron Collisions - 19

  20. The Right Choice • Current Vincia without matching, but with “improved” antenna functions (including suppressed unordered branchings) • Removes dead zone + still better approx than virt-ordered • (Good initial guess  better reweighting efficiency) • Problem 2: leftover Subleading Logs after matching • There are still unsubtractred subleading divergences in the ME A Guide to Hadron Collisions - 20

  21. Matching in Vincia • We are pursuing three strategies in parallel • Addition (aka subtraction) • Simplest & guaranteed to fill all of phase space (unsubtracted ME in dead regions) • But has generic negative weights and hard to exponentiate corrections • Multiplication (aka reweighting) • Guaranteed positive weights & “automatically” exponentiates  path to NLL • Complicated, so 1-loop matching difficult beyond first order. • Only fills phase space populated by shower: dead zones problematic • Hybrid • Combine: simple expansions, full phase space, positive weights, and exponentiation? • Goal • Multi-leg “plug-and-play” NLO + “improved”-LL shower Monte Carlo • Including uncertainty bands (exploring uncontrolled terms) • Extension to NNLO + NLL ? A Guide to Hadron Collisions - 21

  22. NLO with Addition Multiplication at this order  α, β = 0 (POWHEG ) • First Order Shower expansion PS Unitarity of shower  3-parton real = ÷ 2-parton “virtual” • 3-parton real correction (A3 = |M3|2/|M2|2 + 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) A Guide to Hadron Collisions - 22

  23. Matching at Higher Orders  Leftover Subleading Logs • 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 … do it by exponentiated matching Note: more legs  more logs, so ultimately will still need regulator. But try to postpone to NNLL level. A Guide to Hadron Collisions - 23

  24. Z4 Matching by multiplication • 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 A Guide to Hadron Collisions - 24

  25. 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 A Guide to Hadron Collisions - 25

  26. General 2nd Order (& NLL Matching) • Include unitary shower (S) and non-unitary “K-factor” (K) corrections • K: event weight modification (special case: add/subtract events) • Non-unitary  changes normalization (“K” factors) • Non-unitary  does not modify Sudakov  not resummed • Finite corrections can go here ( + regulated logs) • Only needs to be evaluated once per event • S: branching probability modification • Unitary  does not modify normalization • Unitary  modifies Sudakov  resummed • All logs should be here • Needs to be evaluated once for every nested 24 branching (if NLL) • Addition/Subtraction: S = 1, K ≠ 1 • Multiplication/Reweighting: S≠ 1 K = 1 • Hybrid: S = logs K = the rest A Guide to Hadron Collisions - 26

  27. VINCIA in Action • Can vary • evolution variable, kinematics maps, radiation functions, renormalization choice, matching strategy (here just varying splitting functions) • At Pure LL, • can definitely see a non-perturbative correction, but hard to precisely constrain it Giele, Kosower, PS : PRD78(2008)014026 + Les Houches ‘NLM’ 2007 A Guide to Hadron Collisions - 27

  28. VINCIA in Action • Can vary • evolution variable, kinematics maps, radiation functions, renormalization choice, matching strategy (here just varying splitting functions) • At Pure LL, • can definitely see a non-perturbative correction, but hard to precisely constrain it Giele, Kosower, PS : PRD78(2008)014026 + Les Houches ‘NLM’ 2007 A Guide to Hadron Collisions - 28

  29. VINCIA in Action • Can vary • evolution variable, kinematics maps, radiation functions, renormalization choice, matching strategy (here just varying splitting functions) • After 2nd order matching • Non-pert part can be precisely constrained. (will need 2nd order logs as well for full variation) Giele, Kosower, PS : PRD78(2008)014026 + Les Houches ‘NLM’ 2007 A Guide to Hadron Collisions - 29

  30. The next big steps • Z3 at one loop • Opens multi-parton matching at 1 loop • Required piece for NNLO matching • If matching can be exponentiated, opens NLL showers • Work in progress • Write up complete framework for additive matching •  NLO Z3 and NNLO matching within reach • Finish complete framework multiplicative matching … • Complete NLL showers slightly further down the road • Then… • Initial state, masses, polarization, subleading color, unstable particles, … • Also interesting that we can take more differentials than just δμR • Something to be learned here even for estimating fixed-order uncertainties? A Guide to Hadron Collisions - 30