1. Peter Skands Theoretical Physics, Fermilab Eugene, February 2009 Higher Order Aspects of Parton Showers

2. Peter Skands A Guide to Hadron Collisions - 2 Fixed Order (all orders) “Experimental” distribution of observable O in production of X : k : legsℓ : loops {p} : momenta Monte Carlo at Fixed Order High-dimensional problem (phase space) d≥5  Monte Carlo integration Principal virtues 1.Stochastic error O(N -1/2 ) independent of dimension 2.Full (perturbative) quantum treatment at each order 3.(KLN theorem: finite answer at each (complete) order) 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 “Monte Carlo”: N. Metropolis, first Monte Carlo calculation on ENIAC (1948), basic idea goes back to Enrico Fermi

3. Peter Skands A Guide to Hadron Collisions - 3 ►Naively, brems suppressed by α s ~ 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 Bremsstrahlung Example: SUSY @ LHC FIXED ORDER pQCD inclusive X + 1 “jet” inclusive X + 2 “jets” LHC - sps1a - m~600 GeVPlehn, Rainwater, PS PLB645(2007)217 (Computed with SUSY-MadGraph) Cross section for 1 or more 50-GeV jets larger than total σ, obviously non- sensical

4. Peter Skands A Guide to Hadron Collisions - 4 Beyond Fixed Order 1 ►dσ X = … ►dσ X+1 ~ dσ X g 2 2 s ab /(s a1 s 1b ) ds a1 ds 1b ►dσ X+2 ~ dσ X+1 g 2 2 s ab /(s a2 s 2b ) ds a2 ds 2b ►dσ X+3 ~ dσ X+2 g 2 2 s ab /(s a3 s 3b ) ds a3 ds 3b ►But it’s not a parton shower, not yet an “evolution” What’s the total cross section we would calculate from this? σ X;tot = int( dσ X ) + int( dσ X+1 ) + int( dσ X+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) dσXdσX α s ab s ai s ib dσ X+1 dσ X+2 This is an approximation of inifinite- order tree-level cross sections “DLA”

5. Peter Skands A Guide to Hadron Collisions - 5 Beyond Fixed Order 2 ►dσ X = … ►dσ X+1 ~ dσ X g 2 2 s ab /(s a1 s 1b ) ds a1 ds 1b ►dσ X+2 ~ dσ X+1 g 2 2 s ab /(s a2 s 2b ) ds a2 ds 2b ►dσ X+3 ~ dσ X+2 g 2 2 s ab /(s a3 s 3b ) ds a3 ds 3b + Unitarisation: σ tot = int( dσ X )  σ X;excl = σ X - σ X+1 - σ X+2 - … ►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 = σ X+1 (Q) = σ X;incl – σ X;excl (Q) Later, some 3-jets migrate further, etc  σ X+n (Q) = σ X;incl – ∑σ X+m<n;excl (Q) This evolution takes place between two scales, Q in ~ s and Q end = Q had ►σ X;excl = int( dσ X ) - int( dσ X+1,2,3,…;excl ) = int( dσ X ) EXP[ - int( dσ X+1 / dσ X ) ] ►σ X;tot = Sum ( σ X+0,1,2,3,…;excl ) = int( dσ X ) dσXdσX α s ab s ai s ib dσ X+1 dσ X+2 Given a jet definition, an event has either 0, 1, 2, or … jets “DLA”

6. Peter Skands A Guide to Hadron Collisions - 6 LL Shower Monte Carlos ►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 ( σ LO, σ NLO, σ NNLO, σ exp, …)  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 Pure Shower (all orders) O: Observable {p} : momenta w X = |M X | 2 or K|M X | 2 S : Evolution operator Leading Order

7. Peter Skands A Guide to Hadron Collisions - 7 “S” (for Shower) ►Evolution Operator, S (as a function of “time” t=1/Q ) Defined in terms of Δ(t 1,t 2 ) (Sudakov)  The integrated probability the system does not change state between t 1 and t 2  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

8. Peter Skands A Guide to Hadron Collisions - 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”, dΦ n+1 /dΦ n ) The renormalization scheme (vertex-by-vertex argument of α s ) The infrared cutoff contour (hadronization cutoff) Variations  Comprehensive uncertainty estimates (showers with uncertainty bands) Matching to MEs (& N n LL?)  Reduced Dependence (systematic reduction of uncertainty)

9. Peter Skands A Guide to Hadron Collisions - 9 A (complete idiot’s) Solution? ►Combine different starting multiplicites  inclusive sample? ►In practice – Combine 1. [X] ME + showering 2. [X + 1 jet] ME + showering 3. … ►Doesn’t work [X] + shower is inclusive [X+1] + shower is also inclusive X inclusive X+1 inclusive X+2 inclusive ≠ X exclusive X+1 exclusive X+2 inclusive Run generator for X (+ shower) Run generator for X+1 (+ shower) Run generator for … (+ shower) Combine everything into one sample What you get What you want Overlapping “bins”One sample

10. Peter Skands A Guide to Hadron Collisions - 10 The Matching Problem ►[X] ME + shower already contains sing{ [X + n jet] ME } So we really just missed the non-LL bits, not the entire ME! Adding full [X + n jet] ME is 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 : w n = [X + n jet] ME – Shower{w n-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 P n = [X + n jet] ME / Shower{[X+n-1 jet] ME } I call these matching approaches “multiplicative”

11. Peter Skands A Guide to Hadron Collisions - 11 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,... 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

12. Peter Skands A Guide to Hadron Collisions - 12 ►Herwig In dead zone: A i = 0  add events corresponding to unsubtracted |M X+1 | Outside dead zone: reweighted à la Pythia  A i = |M X+1 |   no additive correction necessary ►CKKW and L-CKKW At this order identical to Herwig, with “dead zone” for k T > k Tcut introduced by hand ►MC@NLO In dead zone: identical to Herwig Outside dead zone: A Herwig > |M X+1 |  w X+1 negative  negative weights ►Pythia A i = |M X+1 | over all of phase space  no additive correction necessary ►Powheg At this order identical to Pythia   no negative weights HERWIG TYPE PYTHIA TYPE Matching to X+1: Tree-level

13. Peter Skands A Guide to Hadron Collisions - 13 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 VINCIA ►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 ( μ R = {evolution scale, p T, s, 2-loop, …} ) Infrared cutoff contour (hadronization cutoff)  Same options as for evolution time, but independent of time  universal choice Dipoles (=Antennae, not CS) – a dual description of QCD a b r VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE Giele, Kosower, PS : hep-ph/0707.3652 + Les Houches 2007

14. Peter Skands A Guide to Hadron Collisions - 14Ordering kTkT m2m2 p T (Ariadne) m ant 1-T collinear Phase Space for 2  3 Partitioned-Dipole Dipole-Antenna EgEg Angle soft

15. Peter Skands A Guide to Hadron Collisions - 15 Second Order ►Second Order Shower expansion for 4 partons (assuming first already matched) min # of paths AR p T + AR recoil max # of paths DZ ►Problem 1: dependence on evolution variable Shower is ordered  t 4 integration only up to t 3  2, 1, or 0 allowed “paths” 0 = Dead Zone : not good for reweighting Q E = p T (i,j,k) = m ij m jk /m ijk 0 1 2 3

16. Peter Skands A Guide to Hadron Collisions - 16 Second Order AVERAGEs of Over/Under-counting ►Second Order Shower expansion for 4 partons (assuming first already matched) Define over/under-counting ratio: PS tree / ME tree 0 1 2 3 NB: AVERAGE of R 4 distribution

17. Peter Skands A Guide to Hadron Collisions - 17 Second Order EXTREMA of Over/Under-counting ►Second Order Shower expansion for 4 partons (assuming first already matched) Define over/under-counting ratio: PS tree / ME tree 0 1 2 3 NB: EXTREMA of R 4 distribution (100M points)

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

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

20. Peter Skands A Guide to Hadron Collisions - 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

21. Peter Skands A Guide to Hadron Collisions - 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 ?

22. Peter Skands A Guide to Hadron Collisions - 22 NLO with Addition ►First Order Shower expansion Unitarity of shower  3-parton real = ÷ 2-parton “virtual” ►3-parton real correction ( A 3 = |M 3 | 2 /|M 2 | 2 + finite terms; α, β ) ►2-parton virtual correction (same example) PS Finite terms cancel in 3-parton O Finite terms cancel in 2- parton O (normalization) Multiplication at this order  α, β = 0 (POWHEG )

23. Peter Skands A Guide to Hadron Collisions - 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.

24. Peter Skands A Guide to Hadron Collisions - 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 2 nd order matching term (with 1 st 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

25. Peter Skands A Guide to Hadron Collisions - 25 The Z  3 1-loop term ►Second order matching term for 3 partons ►Additive (S=1)  Ordinary NLO subtraction + shower leftovers Shower off w 2 (V) “Coherence” term: difference between 2- and 3-parton (power-suppressed) evolution above Q E3. Explicit Q E -dependence cancellation. δ α : Difference between alpha used in shower (μ = p T ) and alpha used for matching  Explicit scale choice cancellation Integral over w 4 (R) in IR region still contains NLL divergences  regulate Logs not resummed, so remaining (NLL) logs in w 3 (R) also need to be regulated ►Multiplicative : S = (1+…)  Modified NLO subtraction + shower leftovers A*S contains all logs from tree-level  w 4 (R) finite. Any remaining logs in w 3 (V) cancel against NNLO  NLL resummation if put back in S

26. Peter Skands A Guide to Hadron Collisions - 26 General 2 nd 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

27. Peter Skands A Guide to Hadron Collisions - 27 ►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 VINCIA in Action Giele, Kosower, PS : PRD78(2008)014026 + Les Houches ‘NLM’ 2007

28. Peter Skands A Guide to Hadron Collisions - 28 ►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 VINCIA in Action Giele, Kosower, PS : PRD78(2008)014026 + Les Houches ‘NLM’ 2007

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

30. Peter Skands A Guide to Hadron Collisions - 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 δμ R Something to be learned here even for estimating fixed-order uncertainties?