1. LoopFest VI, Fermilab, April 2007 Parton Showers and NLO Matrix Elements Peter Skands Fermilab / Particle Physics Division / Theoretical Physics In collaboration with W. Giele, D. Kosower

2. Peter SkandsParton Showers and NLO Matrix Elements 2Overview ►Parton Showers QCD & Event Generators Antenna Showers: VINCIA Expansion of the VINCIA shower ►Matching LL shower + tree-level matching (through to α s 2 ) E.g. [X] (0), [X + jet] (0), [X + 2 jets] (0) + shower (~ CKKW, but different) LL shower + 1-loop matching (through to α s ) E.g. [X] (0,1), [X + jet] (0) + shower (~ MC@NLO, but different) A sketch of further developments

3. Peter SkandsParton Showers and NLO Matrix Elements 3 ►Main Tool Approximate by truncation of perturbative series at fixed coupling order Example: Q uantum C hromo D ynamics Reality is more complicated

4. Peter SkandsParton Showers and NLO Matrix Elements 4 Traditional Event Generators ►Basic aim: improve lowest order perturbation theory by including leading corrections  exclusive event samples 1.sequential resonance decays 2.bremsstrahlung 3.underlying event 4.hadronization 5.hadron (and τ ) decays E.g. PYTHIA 2006: first publication of PYTHIA manual JHEP 0605:026,2006 (FERMILAB-PUB-06-052-CD-T)

5. Peter SkandsParton Showers and NLO Matrix Elements 5 T he B ottom L ine The S matrix is expressible as a series in g i, g i n /t m, g i n /x m, g i n /m m, g i n /f π m, … 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 FODGLAP BFKL HQET χPT

6. Peter SkandsParton Showers and NLO Matrix Elements 6 Improved Parton Showers ►Step 1: A comprehensive look at the uncertainty (here PS @ LL) Vary the evolution variable (~ factorization scheme) Vary the antenna function Vary the kinematics map (angle around axis perp to 2  3 plane in CM) Vary the renormalization scheme (argument of α s ) Vary the infrared cutoff contour (hadronization cutoff) ►Step 2: Systematically improve on it Understand how each variation could be cancelled when Matching to fixed order matrix elements Higher logarithms are included ►Step 3: Write a generator Make the above explicit (while still tractable) in a Markov Chain context  matched parton shower MC algorithm Subject of this talk

7. Peter SkandsParton Showers and NLO Matrix Elements 7 VINCIA ►VINCIA Dipole shower C++ code for gluon showers Standalone since ~ half a year Plug-in to PYTHIA 8 (C++ PYTHIA) since ~ last week Most results presented here use the plug-in version ►So far: 2 different shower evolution variables: pT-ordering (~ ARIADNE, PYTHIA 8) Virtuality-ordering (~ PYTHIA 6, SHERPA) For each: an infinite family of antenna functions shower functions = leading singularities plus arbitrary polynomials (up to 2 nd order in s ij ) Shower cutoff contour: independent of evolution variable  IR factorization “universal”  less wriggle room for non-pert physics? Phase space mappings: 3 choices implemented ARIADNE angle, Emitter + Recoiler, or “DK1” (+ ultimately smooth interpolation?) Dipoles – a dual description of QCD 1 3 2 virtual numerical collider with interesting antennae Giele, Kosower, PS : in progress

8. Peter SkandsParton Showers and NLO Matrix Elements 8 Checks: Analytic vs Numerical vs Splines ►Calculational methods 1.Analytic integration over resolved region (as defined by evolution variable) – obtained by hand, used for speed and cross checks 2.Numeric: antenna function integrated directly (by nested adaptive gaussian quadrature)  can put in any function you like 3.In both cases, the generator constructs a set of natural cubic splines of the given Sudakov (divided into 3 regions linearly in Q R – coarse, fine, ultrafine) ►Test example Precision target: 10 -6 gg  ggg Sudakov factor (with nominal α s = unity) gg  ggg: Δ(s,Q 2 ) Analytic Splined p T -ordered Sudakov factor Numeric / Analytic Spline (3x1000 points) / Analytic Ratios Spline off by a few per mille at scales corresponding to less than a per mille of all dipoles  global precision ok ~ 10 -6 VINCIA 0.010 (Pythia8 plug-in version) (a few experiments with single & double logarithmic splines  no huge success. So far linear ones ok for desired speed & precision)

9. Peter SkandsParton Showers and NLO Matrix Elements 9 Why Splines? ►Example: m H = 120 GeV H  gg + shower Shower start: 120 GeV. Cutoff = 1 GeV ►Speed (2.33 GHz, g++ on cygwin) Tradeoff: small downpayment at initialization  huge interests later &v.v. (If you have analytic integrals, that’s great, but must be hand-made) Aim to eventually handle any function & region  numeric more general Initialization (PYTHIA 8 + VINCIA) 1 event Analytic, no splines2s(< 10 -3 s ?) Analytic + splines2s< 10 -3 s Numeric, no splines2s6s Numeric + splines50s< 10 -3 s Numerically integrate the antenna function (= branching probability) over the resolved 2D branching phase space for every single Sudakov trial evaluation Have to do it only once for each spline point during initialization

10. Peter SkandsParton Showers and NLO Matrix Elements 10Matching ►Matching of up to one hard additional jet PYTHIA-style (reweight shower: ME = w*PS) HERWIG-style (add separate events from ME: weight = ME-PS) MC@NLO-style (ME-PS subtraction similar to HERWIG, but NLO) ►Matching of generic (multijet) topologies (at tree level) ALPGEN-style (MLM) SHERPA-style (CKKW) ARIADNE-style (Lönnblad-CKKW) PATRIOT-style (Mrenna & Richardson) ►Brand new approaches (still in the oven) Refinements of MC@NLO (Nason) CKKW-style at NLO (Nagy, Soper) SCET approach (based on SCET – Bauer, Schwarz) VINCIA (based on QCD antennae – Giele, Kosower, PS) Evolution

11. Peter SkandsParton Showers and NLO Matrix Elements 11MC@NLO Nason’s approach: Generate 1 st shower emission separately  easier matching Avoid negative weights + explicit study of ZZ production Frixione, Nason, Webber, JHEP 0206(2002)029 and 0308(2003)007 JHEP 0411(2004)040 JHEP 0608(2006)077 ►MC@NLO in comparison Superior precision for total cross section Equivalent to tree-level matching for event shapes (differences higher order) Inferior to multi-jet matching for multijet topologies So far has been using HERWIG parton shower  complicated subtractions

12. Peter SkandsParton Showers and NLO Matrix Elements 12 Expanding the Shower ►Start from Sudakov factor = No-branching probability: (n or more  n and only n) ►Decompose inclusive cross section ►Simple example (sufficient for matching through NLO): NB: simplified notation! Differentials are over entire respective phase spaces Sums run over all possible branchings of all antennae

13. Peter SkandsParton Showers and NLO Matrix Elements 13 Matching at NLO: tree part ►NLO real radiaton term from parton shower ►Add extra tree-level X + jet (at this point arbitrary) ►Correction term is given by matching to fixed order:  variations (or dead regions) in |a| 2 canceled by matching at this order (If |a| too hard, correction can become negative  constraint on |a|) ►Subtraction can be automated from ordinary tree-level ME’s + no dependence on unphysical cut or preclustering scheme (cf. CKKW) - not a complete order: normalization changes (by integral of correction), but still LO NB: simplified notation! Differentials are over entire respective phase spaces Sums run over all possible branchings of all antennae Twiddles = finite (subtracted) ME corrections Untwiddled = divergent (unsubtracted) MEs

14. Peter SkandsParton Showers and NLO Matrix Elements 14 Matching at NLO: loop part ►NLO virtual correction term from parton shower ►Add extra finite correction (at this point arbitrary) ►Have to be slightly more careful with matching condition (include unresolved real radiation) but otherwise same as before: ►Probably more difficult to fully automate, but |a| 2 not shower-specific Currently using Gehrmann-Glover (global) antenna functions Will include also Kosower’s (sector) antenna functions Tree-level matching just corresponds to using zero (This time, too small |a|  correction negative)

15. Peter SkandsParton Showers and NLO Matrix Elements 15 Matching at NNLO: tree part ►Adding more tree-level MEs is straightforward ►Example: second emission term from NLO matched parton shower ►X+2 jet tree-level ME correction term and matching equation Matching equation looks identical to 2 slides ago If all indices had been shown: sub-leading colour structures not derivable by nested 2  3 branchings do not get subtracted

16. Peter SkandsParton Showers and NLO Matrix Elements 16 Matching at NNLO: tree part, with 2  4 ►Sketch only! But from matching point of view at least, no problem to include 2  4 ►Second emission term from NLO matched parton shower with 2  4 (For subleading colour structures, only |b| 2 term enters) ►Correction term and matching equation (Again, for subleading colour structures, only |b| 2 term is non-zero) ►So far showing just for fun (and illustration) Fine that matching seems to be ok with it, but … Need complete NLL shower formalism to resum 2  4 consistently If possible, would open the door to MC@NNLO

17. Peter SkandsParton Showers and NLO Matrix Elements 17 Under the Rug ►The simplified notation allowed to skip over a few issues we want to understand slightly better, many of them related Start and re-start scales for the shower away from the collinear limit Evolution variable: global vs local definitions How the arbitrariness in the choice of phase space mapping is canceled by matching How the arbitrariness in the choice of evolution variable is canceled by matching Constructing an exactly invertible shower (sector decomposition) Matching in the presence of a running renormalization scale Dependence on the infrared factorization (hadronization cutoff) Degree of automation and integration with existing packages To what extent negative weights (oversubtraction) may be an issue ►None of these are showstoppers as far as we can tell

18. Peter SkandsParton Showers and NLO Matrix Elements 18 Under the Rug 2 ►I explained the method in some detail in order not to have much time left at this point ►We are now concentrating on completing the shower part for Higgs decays to gluons, so no detailed pheno studies yet The aim is to get a standalone paper on the shower out faster, accompanied by the shower plug-in for PYTHIA 8 We will then follow up with a writeup on the matching ►I will just show an example based on tree-level matching for H  gg

19. Peter SkandsParton Showers and NLO Matrix Elements 19 VINCIA Example: H  gg  ggg VINCIA 0.008 Unmatched “soft” |A| 2 VINCIA 0.008 Unmatched “hard” |A| 2 VINCIA 0.008 Matched “soft” |A| 2 VINCIA 0.008 Matched “hard” |A| 2 y 12 y 23 y 12 ►First Branching ~ first order in perturbation theory ►Unmatched shower varied from “soft” to “hard” : soft shower has “radiation hole”. Filled in by matching. radiation hole in high-p T region Outlook: Immediate Future: Paper about gluon shower Include quarks  Z decays Matching Then: Initial State Radiation Hadron collider applications

20. Peter SkandsParton Showers and NLO Matrix Elements 20 A Problem ►The best of both worlds? We want: A description which accurately predicts hard additional jets + jet structure and the effects of multiple soft emissions ►How to do it? Compute emission rates by parton showering? Misses relevant terms for hard jets, rates only correct for strongly ordered emissions p T1 >> p T2 >> p T3... (common misconception that showers are soft, but that need not be the case. They can err on either side of the right answer.) Unknown contributions from higher logarithmic orders Compute emission rates with matrix elements? Misses relevant terms for soft/collinear emissions, rates only correct for well-separated individual partons Quickly becomes intractable beyond one loop and a handfull of legs Unknown contributions from higher fixed orders

21. Peter SkandsParton Showers and NLO Matrix Elements 21 Double Counting ►Combine different multiplicites  inclusive sample? ►In practice – Combine 1.[X] ME + showering 2.[X + 1 jet] ME + showering 3.… ►  Double Counting: [X] ME + showering produces some X + jet configurations The result is X + jet in the shower approximation If we now add the complete [X + jet] ME as well the total rate of X+jet is now approximate + exact ~ double !! some configurations are generated twice. and the total inclusive cross section is also not well defined ►When going to X, X+j, X+2j, X+3j, etc, this problem gets worse  X inclusive X+1 inclusive X+2 inclusive ≠ X exclusive X+1 exclusive X+2 inclusive

22. Peter SkandsParton Showers and NLO Matrix Elements 22 The simplest example: ALPGEN ►“MLM” matching (proposed by Michelangelo “L” Mangano) Simpler but similar in spirit to “CKKW” ►First generate events the “stupid” way: 1.[X n ] ME + showering 2.[X n+1 ] ME + showering 3.… ►  A set of fully showered events, with double counting. To get rid of the excess, accept/reject each event based on: (cone-)cluster showered event  n jets Check each parton from the Feynman diagram  one jet? If all partons are ‘matched’, keep event. Else discard it. ►Virtue: can be done without knowledge of the internal workings of the generator. Only the fully showered final events are needed  Simple procedure to improve multijet rates in perturbative QCD n inclusive n+1 inclusive n+2 inclusive n exclusive n+1 exclusive n+2 inclusive