1. GGI October 2007 Red, Blue, and Green Things With Antennae Peter Skands CERN & Fermilab In collaboration with W. Giele, D. Kosower Giele, Kosower, PS : hep-ph/0707.3652 ; Sjöstrand, Mrenna, PS : hep-ph/0710.3820)

2. Peter SkandsRed, Blue, and Green Things with Antennae 2Aims ►We’d like a simple formalism for parton showers that allows: 1.Extensive analytical control, including systematic uncertainty estimates 2.Combining the virtues of CKKW (Tree-level matching with arbitrarily many partons) with those of MC@NLO (Loop-level matching) 3.To eventually go beyond Leading-NC+Leading-Log? ►Two central ingredients A general QCD cascade based on exponentiated antennae  a generalized “ARIADNE” Extending the ideas of subtraction-based ME/PS matching  “MC@NLO” with antennae Plus works for more than 1 hard leg  Simultaneous matching to several tree- or 1- loop matrix elements

3. Peter SkandsRed, Blue, and Green Things with Antennae 3 Monte Carlo Basics High-dimensional problem (phase space)  Monte Carlo integration + Formulation of fragmentation as a “Markov Chain”: 1.Parton Showers: iterative application of perturbatively calculable splitting kernels for n  n+1 partons ( = resummation of soft/collinear logarithms)  Ordered evolution. 2.Hadronization: iteration of X  X + hadron, at present according to phenomenological models, based on known properties of QCD, on lattice, and on fits to data. Principal virtues 1.Stochastic error O(N -1/2 ) independent of dimension 2.Universally applicable 3.Fully exclusive final states (for better or for worse – cf. the name ‘Pythia’ … ) 4.  no separate calculation needed for each observable 5.Have become indispensable for experimental studies. Exclusivity  possible to calculate a detector response event by event.

4. Peter SkandsRed, Blue, and Green Things with Antennae 4 ►Iterative (Markov chain) formulation = parton shower Generates leading “soft/collinear” corrections to any process, to infinite order in the coupling The chain is ordered in an “evolution variable”: e.g. parton virtuality, jet-jet angle, transverse momentum, …  a series of successive factorizations the lower end of which can be matched to a hadronization description at some fixed low hadronization scale ~ 1 GeVBasics

5. Peter SkandsRed, Blue, and Green Things with Antennae 5 Improved Event Generators ►Step 1: A comprehensive look at the uncertainty Vary the evolution variable (~ factorization scheme) Vary the radiation function (finite terms not fixed) 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 it Understand the importance of each and how it is canceled by Matching to fixed order matrix elements Higher logarithms, subleading color, etc, 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

6. Peter SkandsRed, Blue, and Green Things with Antennae 6 Example: Z decays ►Dependence on evolution variable

7. Peter SkandsRed, Blue, and Green Things with Antennae 7 Example: Z decays ►Dependence on evolution variable Using α s (p T ), p Thad = 0.5 GeV α s (m Z ) = 0.137 N f = 2 for all plots Note: the default Vincia antenna functions reproduce the Z  3 parton matrix element; Pythia8 includes matching to Z  3 Beyond the 3 rd parton, Pythia’s radiation function is slightly larger, and its kinematics and hadronization cutoff contour are also slightly different

8. Peter SkandsRed, Blue, and Green Things with Antennae 8 The main sources of uncertainty ►Things I can tell you about: Non-singular terms in the radiation functions The choice of renormalization scale (The hadronization cutoff) ►Things I can’t (yet) tell you about Effects of sub-leading logarithms Effects of sub-leading colours Polarization effects

9. Peter SkandsRed, Blue, and Green Things with Antennae 9 VINCIA ►VINCIA shower Final-state QCD cascades At GGI  completed (massless) quarks Plug-in to PYTHIA 8.1 (C++) + working on initial state … ►So far: 2 different shower evolution variables: pT-ordering (~ ARIADNE, PYTHIA 8) Mass-ordering (~ PYTHIA 6, SHERPA) For each: an infinite family of antenna functions Laurent series in the branching invariants: singular part fixed by QCD, finite terms arbitrary Shower cutoff contour: independent of evolution variable  IR factorization “universal” Phase space mappings: 2 different choices implemented Antenna-like (ARIADNE angle) or Parton-shower-like: Emitter + longitudinal Recoiler Dipoles – a dual description of QCD 1 3 2 VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE Giele, Kosower, PS : hep-ph/0707.3652 Gustafson, Phys. Lett. B175 (1986) 453 Lönnblad, Comput. Phys. Commun. 71 (1992) 15.

10. Peter SkandsRed, Blue, and Green Things with Antennae 10 The Pure Shower Chain ►Shower-improved distribution of an observable: ►Shower Operator, S (as a function of “time” t=1/Q ) ►n-parton Sudakov ►Focus on antenna-dipole showers Dipole branching phase space “X + nothing” “X+something” Giele, Kosower, PS : hep-ph/0707.3652

11. Peter SkandsRed, Blue, and Green Things with Antennae 11 Dipole-Antenna Functions ►Starting point: “GGG” antenna functions, e.g., ►Generalize to arbitrary Laurent series (at GGI): ►  Can make shower systematically “softer” or “harder” Will see later how this variation is explicitly canceled by matching  quantification of uncertainty  quantification of improvement by matching (In principle, could also “fake” other showers) y ar = s ar / s i s i = invariant mass of i’th dipole-antenna Giele, Kosower, PS : hep-ph/0707.3652 Gehrmann-De Ridder, Gehrmann, Glover, JHEP 09 (2005) 056 Singular parts fixed, finite terms arbitrary

12. Peter SkandsRed, Blue, and Green Things with Antennae 12Matching Fixed Order (all orders) Matched shower (including simultaneous tree- and 1-loop matching for any number of legs) Tree-level “real” matching term for X+k Loop-level “virtual” matching term for X+k Pure Shower (all orders)

13. Peter SkandsRed, Blue, and Green Things with Antennae 13 Tree-level matching to X+1 ►First order real radiation term from parton shower ►Matrix Element (X+1 at LO ; above t had )  Matching Term:  variations in finite terms (or dead regions) in A canceled by matching at this order (If A too hard, correction can become negative  negative weights) ►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 Inverse kinematics map = clustering

14. Peter SkandsRed, Blue, and Green Things with Antennae 14 1-loop matching to X ►NLO “virtual term” from parton shower (= expanded Sudakov: exp=1 - … ) ►Matrix Elements (unresolved real plus genuine virtual) ►Matching condition same as before (almost): ►May be automated ?, anyway: A is not shower-specific Currently using Gehrmann-Glover (global) antenna functions You can choose anything as long as you can write it as a Laurent series Tree-level matching just corresponds to using zero (This time, too small A  correction negative)

15. Peter SkandsRed, Blue, and Green Things with Antennae 15

16. Peter SkandsRed, Blue, and Green Things with Antennae 16 Fromto ! ►The unknown finite terms are a major source of uncertainty DGLAP has some, GGG have others, ARIADNE has yet others, etc… They are arbitrary (and in general process-dependent) Using α s (M Z )=0.137, μ R =1/4m dipole, p Thad = 0.5 GeV

17. Peter SkandsRed, Blue, and Green Things with Antennae 17 Tree-level matching to X+1 ►First order real radiation term from parton shower ►Matrix Element (X+1 at LO ; above t had )  Matching Term:  variations in finite terms (or dead regions) in A canceled by matching at this order (If A too hard, correction can become negative  negative weights)

18. Peter SkandsRed, Blue, and Green Things with Antennae 18 Phase Space Population SoftStandardHard Matched SoftStandardMatched Hard Positive correctionNegative correction

19. Peter SkandsRed, Blue, and Green Things with Antennae 19 Fromto ! ►The unknown finite terms are a major source of uncertainty DGLAP has some, GGG have others, ARIADNE has yet others, etc… They are arbitrary (and in general process-dependent) Using α s (M Z )=0.137, μ R =1/4m dipole, p Thad = 0.5 GeV

20. Peter SkandsRed, Blue, and Green Things with Antennae 20 Note about “NLO” matching ►Shower off virtual matching term  uncanceled O(α 2 ) contribution to 3-jet observables (only canceled by 1-loop 3-parton matching) ►While normalization is improved, shapes are not (shape still LO) Using α s (M Z )=0.137, μ R =1/4m dipole, p Thad = 0.5 GeV

21. Peter SkandsRed, Blue, and Green Things with Antennae 21 What to do next? ►Go further with tree-level matching Demonstrate it beyond first order (include H,Z  4 partons) May require “true Markov” evolution and/or “sector” antenna functions? Automated tree-level matching (w. Rikkert Frederix (MadGraph) + …?) ►Go further with one-loop matching Demonstrate it beyond first order (include 1-loop H,Z  3 partons) Should start to see cancellation of ordering variable and renormalization scale Should start to see stabilization of shapes as well as normalizations ►Extend the formalism to the initial state ►Extend to massive particles Massive antenna functions, phase space, and evolution ►Investigate possibilities for going beyond traditional LL showers: Subleading colour, NLL, and polarization

22. Peter SkandsRed, Blue, and Green Things with Antennae 22 Last Slide Thanks to the organizers A. Brandhuber (Queen Mary University, London),A. Brandhuber (Queen Mary University, London), V. Del Duca (INFN, Torino),V. Del Duca (INFN, Torino), N. Glover (IPPP, Durham),N. Glover (IPPP, Durham), D. Kosower (CEA, Saclay),D. Kosower (CEA, Saclay), E. Laenen (Nikhef, Amsterdam),E. Laenen (Nikhef, Amsterdam), G. Passarino (University of Torino),G. Passarino (University of Torino), W. Spence (Queen Mary University, London),W. Spence (Queen Mary University, London), G. Travaglini (Queen Mary University, London),G. Travaglini (Queen Mary University, London), D. Zeppenfeld (University of Karlsruhe)D. Zeppenfeld (University of Karlsruhe)