Parton Showers and NLO Matrix Elements: Improved Shower Algorithm

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

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

3. Improved Parton Showers • Step 1: A comprehensive look at the uncertainty (here PS @ 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 αs) • 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 Parton Showers and NLO Matrix Elements

4. The Pure Shower Chain “X + nothing” “X+something” Dipole branching phase space Giele, Kosower, PS : FERMILAB-PUB-07-160-T • 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 Parton Showers and NLO Matrix Elements

5. 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 2nd order in sij) 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?) VINCIA VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE Giele, Kosower, PS : FERMILAB-PUB-07-160-T Gustafson, Phys. Lett. B175 (1986) 453 1 Dipoles – a dual description of QCD 2 3 Lonnblad, Comput. Phys. Commun. 71 (1992) 15. Parton Showers and NLO Matrix Elements

6. Dipole-Antenna Functions Giele, Kosower, PS : FERMILAB-PUB-07-160-T • 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 Gehrmann-De Ridder, Gehrmann, Glover, JHEP 09 (2005) 056 yar = sar / si si = invariant mass of i’th dipole-antenna Parton Showers and NLO Matrix Elements

7. Checks: Analytic vs Numerical vs Splines • Calculational methods • Analytic integration over resolved region (as defined by evolution variable) – obtained by hand, used for speed and cross checks • Numeric: antenna function integrated directly (by nested adaptive gaussian quadrature)  can put in any function you like • In both cases, the generator constructs a set of natural cubic splines of the given Sudakov (divided into 3 regions linearly in QR – coarse, fine, ultrafine) • Test example • Precision target: 10-6 • ggggg Sudakov factor (with nominal αs= unity) pT-ordered Sudakov factor • ggggg: Δ(s,Q2) • Analytic • Splined VINCIA 0.010 (Pythia8 plug-in version) 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 Numeric / Analytic Spline (3x1000 points) / Analytic (a few experiments with single & double logarithmic splines  not huge success. So far linear ones ok for desired speed & precision) Parton Showers and NLO Matrix Elements

8. Why Splines? Numerically integrate the antenna function (= branching probability) over the resolved 2D branching phase space for every single Sudakov trial evaluation • Example: mH = 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 Have to do it only once for each spline point during initialization Parton Showers and NLO Matrix Elements

9. Matching • “X matched to n resolved partons at leading order and m < n at next-to-leading order” should fulfill Resolved = with respect to the infrared (hadronization) shower cutoff Giele, Kosower, PS : FERMILAB-PUB-07-160-T Fixed Order Matched shower (NLO) LO matching term for X+k NLO matching term for X+k Parton Showers and NLO Matrix Elements

10. Matching to X+1 at LO Giele, Kosower, PS : FERMILAB-PUB-07-160-T • First order real radiation term from parton shower • Matrix Element (X+1 at LO ; above thad) •  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 Parton Showers and NLO Matrix Elements

11. Matching to X at NLO Giele, Kosower, PS : FERMILAB-PUB-07-160-T • 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) Parton Showers and NLO Matrix Elements

12. Matching to X+2 at LO Giele, Kosower, PS : FERMILAB-PUB-07-160-T • 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 pT 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 Parton Showers and NLO Matrix Elements

13. Going deeper? Giele, Kosower, PS : FERMILAB-PUB-07-160-T • 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 Parton Showers and NLO Matrix Elements

14. Universal Hadronization Giele, Kosower, PS : FERMILAB-PUB-07-160-T • 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 pT-ordered shower along a contour of constant m2 • 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 Parton Showers and NLO Matrix Elements

15. “Sudakov” vs LUCLUS pT Giele, Kosower, PS : FERMILAB-PUB-07-160-T 2-jet rate vs PYCLUS pT (= LUCLUS ~ JADE) Preliminary! Vincia “hard” & “soft” Vincia nominal Pythia8 Same variations Parton Showers and NLO Matrix Elements

16. VINCIA Example: H  gg  ggg Giele, Kosower, PS : FERMILAB-PUB-07-160-T • First Branching ~ first order in perturbation theory • Unmatched shower varied from “soft” to “hard” : soft shower has “radiation hole”. Filled in by matching. • Outlook: • Immediate Future: • Paper about gluon shower • Include quarks  Z decays • Automated matching • Then: • Initial State Radiation • Hadron collider applications VINCIA 0.008 Unmatched “soft” |A|2 VINCIA 0.008 Matched “soft” |A|2 y23 y23 radiation hole in high-pT region y23 y23 VINCIA 0.008 Unmatched “hard” |A|2 VINCIA 0.008 Matched “hard” |A|2 y12 y12 Parton Showers and NLO Matrix Elements