Les Houches 2007 VINCIA Peter Skands Fermilab / Particle Physics Division / Theoretical Physics In collaboration with W. Giele, D. Kosower

Peter SkandsParton Showers and NLO Matrix Elements 2Aims ►We’d like a simple formalism for parton showers that allows: 1.Including systematic uncertainty estimates 2.Combining the virtues of CKKW (LO matching with arbitrarily many partons) with those of (NLO matching) ►We have done this by expanding on the ideas of Frixione, Nason, and Webber but with a few substantial generalizations

Peter SkandsParton Showers and NLO Matrix Elements 3 Improved Parton Showers ►Step 1: A comprehensive look at the uncertainty (here LL) 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 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

Peter SkandsParton Showers and NLO Matrix Elements 4 The Pure Shower Chain ►Shower-improved (= resummed) distribution of an observable: ►Shower Operator, S (as a function of (invariant) “time” t=1/Q ) ►n-parton Sudakov ►Focus on dipole showers Dipole branching phase space “X + nothing” “X+something” Giele, Kosower, PS : FERMILAB-PUB T

Peter SkandsParton Showers and NLO Matrix Elements 5 VINCIA ►VINCIA Dipole shower C++ code for gluon showers Standalone since ~ half a year Plug-in to PYTHIA 8 (C++ PYTHIA) since ~ a month 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” Phase space mappings: 3 different choices implemented ARIADNE angle, Emitter + Recoiler, or “DAK” (+ ultimately smooth interpolation?) Dipoles – a dual description of QCD VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE Giele, Kosower, PS : FERMILAB-PUB T Gustafson, Phys. Lett. B175 (1986) 453 Lonnblad, Comput. Phys. Commun. 71 (1992) 15.

Peter SkandsParton Showers and NLO Matrix Elements 6 Dipole-Antenna Functions ►Starting point: de-Ridder-Gehrmann-Glover ggg antenna functions ►Generalize to arbitrary finite terms: ►  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 y ar = s ar / s i s i = invariant mass of i’th dipole-antenna Giele, Kosower, PS : FERMILAB-PUB T Gehrmann-De Ridder, Gehrmann, Glover, JHEP 09 (2005) 056

Peter SkandsParton Showers and NLO Matrix Elements 7 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: 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 ~ VINCIA (Pythia8 plug-in version) (a few experiments with single & double logarithmic splines  not huge success. So far linear ones ok for desired speed & precision)

Peter SkandsParton Showers and NLO Matrix Elements 8 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 interest 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(< s ?) Analytic + splines2s< s Numeric, no splines2s6s Numeric + splines50s< 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

Peter SkandsParton Showers and NLO Matrix Elements 9Matching ►“X matched to n resolved partons at leading order and m < n at next-to-leading order” should fulfill Fixed Order Matched shower (NLO) Resolved = with respect to the infrared (hadronization) shower cutoff LO matching term for X+k NLO matching term for X+k Giele, Kosower, PS : FERMILAB-PUB T

Peter SkandsParton Showers and NLO Matrix Elements 10 Matching to X+1 at LO ►First order real radiation term from parton shower ►Matrix Element (X+1 at LO ; above t had ) ►  Matching Term:  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 Giele, Kosower, PS : FERMILAB-PUB T

Peter SkandsParton Showers and NLO Matrix Elements 11 Matching to X at NLO ►NLO “virtual term” from parton shower (= expanded Sudakov: exp=1 - … ) ►Matrix Element ►Have to be slightly more careful with matching condition (include unresolved real radiation) but otherwise same as before: ►May be automated using complex momenta, and |a| 2 not shower-specific Currently using Gehrmann-Glover (global) antenna functions Will include also Kosower’s (sector) antenna functions (only ever one dipole contributing to each PS point  shower unique and exactly invertible) Tree-level matching just corresponds to using zero (This time, too small |a|  correction negative) Giele, Kosower, PS : FERMILAB-PUB T

Peter SkandsParton Showers and NLO Matrix Elements 12 ? Matching to X+2 at LO ►Adding more tree-level MEs is (pretty) straightforward Example: second emission term from NLO matched parton shower ►Must be slightly careful: unsubtracted subleading logs be here Formally subtract them? Cut them out with a p T cut? Smooth alternative: kill them using the Sudakov? But note: this effect is explicitly NLL (cf. CKKW) 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 Giele, Kosower, PS : FERMILAB-PUB T

Peter SkandsParton Showers and NLO Matrix Elements 13 Going deeper? ►NLL Sudakov with 2  4 B terms should be LL subtracted (LL matched) to avoid double counting ►No problem from matching point of view: ►Could also imagine: higher-order coherence by higher multipoles 6D branching phase space = more tricky Giele, Kosower, PS : FERMILAB-PUB T

Peter SkandsParton Showers and NLO Matrix Elements 14 Universal Hadronization ►Sometimes talk about “plug-and-play” hadronization This generally leads to combinations of frowns and ticks: showers are (currently) intimately tied to their hadronization models, fitted together  Liberate them ►Choose IR shower cutoff (hadronization cutoff) to be universal and independent of the shower evolution variable E.g. cut off a p T -ordered shower along a contour of constant m 2 This cutoff should be perceived as part of the hadronization model. Can now apply the same hadronization model to another shower Good up to perturbative ambiguities Especially useful if you have several infinite families of parton showers Giele, Kosower, PS : FERMILAB-PUB T

Peter SkandsParton Showers and NLO Matrix Elements 15 “Sudakov” vs LUCLUS p T Giele, Kosower, PS : FERMILAB-PUB T Vincia “hard” & “soft” Vincia nominal Pythia8 Same variations 2-jet rate vs PYCLUS p T (= LUCLUS ~ JADE) Preliminary!

Peter SkandsParton Showers and NLO Matrix Elements 16 VINCIA Example: H  gg  ggg VINCIA Unmatched “soft” |A| 2 VINCIA Unmatched “hard” |A| 2 VINCIA Matched “soft” |A| 2 VINCIA 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 Automated matching Then: Initial State Radiation Hadron collider applications Giele, Kosower, PS : FERMILAB-PUB T