Loop Fest VI Fermilab April 2007 Parton Showers
Loop. Fest 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
Overview ► Parton Showers • QCD & Event Generators • Antenna Showers: VINCIA • Expansion of the VINCIA shower ► Matching • LL shower + tree-level matching (through to αs 2) • E. g. [X](0) , [X + jet](0) , [X + 2 jets](0) + shower (~ CKKW, but different) • LL shower + 1 -loop matching (through to αs) • E. g. [X](0, 1) , [X + jet](0) + shower (~ MC@NLO, but different) • A sketch of further developments Peter Skands Parton Showers and NLO Matrix Elements 2
Q C D uantum hromo ynamics ► Main Tool • Approximate by truncation of perturbative series at fixed coupling order • Example: Reality is more complicated Peter Skands Parton Showers and NLO Matrix Elements 3
Traditional Event Generators ► Basic aim: improve lowest order perturbation theory by including leading corrections exclusive event samples 1. 2. 3. 4. 5. Peter Skands sequential resonance decays bremsstrahlung underlying event hadronization hadron (and τ) decays E. g. PYTHIA 2006: first publication of PYTHIA manual JHEP 0605: 026, 2006 (FERMILAB-PUB-06 -052 -CD-T) Parton Showers and NLO Matrix Elements 4
T B L he ottom FO ine HQET DGLAP The S matrix is expressible as a series in gi, gin/tm, gin/xm, gin/mm, gin/fπm , … To do precision physics: BFKL χPT 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 Peter Skands Parton Showers and NLO Matrix Elements 5
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 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 Subject of this talk ► Step 3: Write a generator • Make the above explicit (while still tractable) in a Markov Chain context matched parton shower MC algorithm Peter Skands Parton Showers and NLO Matrix Elements 6
VINCIA VIRTUAL NUMERICAL COLLIDER WITH INTERACTING ANTENNAE Giele, Kosower, PS : in progress ► VINCIA Dipole shower • C++ code for gluon showers Dipoles – a dual description of QCD 1 • Standalone since ~ half a year • Plug-in to PYTHIA 8 (C++ PYTHIA) since ~ last week • Most results presented here use the plug-in version 2 ► So far: • 2 different shower evolution variables: • p. T-ordering (~ ARIADNE, PYTHIA 8) • Virtuality-ordering (~ PYTHIA 6, SHERPA) 3 • For each: an infinite family of antenna functions • shower functions = leading singularities plus arbitrary polynomials (up to 2 nd order in sij) • 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 “DK 1” (+ ultimately smooth interpolation? ) Peter Skands Parton 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) – p. T-ordered Sudakov factor gg ggg: Δ(s, Q 2) • Analytic • Splined obtained by hand, used for speed and cross checks 2. Numeric: antenna function integrated directly (by nested adaptive gaussian quadrature) can put in any VINCIA 0. 010 (Pythia 8 plug-in version) function you like 3. In both cases, the generator constructs a set of natural cubic splines of the given Sudakov (divided into 3 regions 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 linearly in QR – coarse, fine, ultrafine) Numeric / Analytic Spline (3 x 1000 points) / Analytic ► Test example • Precision target: 10 -6 • gg ggg Sudakov factor (with nominal αs = unity) Peter Skands (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 2 D branching phase space for every single Sudakov trial evaluation ► Example: m. H = 120 Ge. V • H gg + shower • Shower start: 120 Ge. V. Cutoff = 1 Ge. V ► Speed (2. 33 GHz, g++ on cygwin) Initialization Have to do it only once for each spline point during initialization 1 event (PYTHIA 8 + VINCIA) Analytic, no splines 2 s (< 10 -3 s ? ) Analytic + splines 2 s < 10 -3 s Numeric, no splines 2 s 6 s 50 s < 10 -3 s Numeric + splines • 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 Peter Skands Parton Showers and NLO Matrix Elements 9
Matching ► 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) Peter Skands • 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) Parton Showers and NLO Matrix Elements Evolution 10
MC@NLO Frixione, Nason, Webber, JHEP 0206(2002)029 and 0308(2003)007 Nason’s approach: JHEP 0411(2004)040 Generate 1 st shower emission separately easier matching ► MC@NLO in comparison • • Peter Skands JHEP 0608(2006)077 Avoid negative weights + explicit study of ZZ production 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 Parton Showers and NLO Matrix Elements 11
Expanding the Shower ► Start from Sudakov factor = No-branching probability: (n or more n and only n) ► Decompose inclusive cross section NB: simplified notation! Differentials are over entire respective phase spaces Sums run over all possible branchings of all antennae ► Simple example (sufficient for matching through NLO): Peter Skands Parton Showers and NLO Matrix Elements 12
Matching at NLO: tree part ► NLO real radiation term from parton shower NB: simplified notation! ► Add extra tree-level X + jet (at this point arbitrary) Differentials are over entire respective phase spaces Sums run over all possible branchings of all antennae ► Correction term is given by matching to fixed order: Twiddles = finite (subtracted) ME corrections Untwiddled = divergent (unsubtracted) MEs • 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 Peter Skands Parton Showers and NLO Matrix Elements 13
Matching at NLO: loop part ► NLO virtual correction term from parton shower ► Add extra finite correction (at this point arbitrary) Tree-level matching just corresponds to using zero ► Have to be slightly more careful with matching condition (include unresolved real radiation) but otherwise same as before: • (This time, too small |a| correction negative) ► Probably more difficult to fully automate, but |a|2 not shower-specific • • Peter Skands Currently using Gehrmann-Glover (global) antenna functions Will include also Kosower’s (sector) antenna functions Parton Showers and NLO Matrix Elements 14
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 Peter Skands Parton Showers and NLO Matrix Elements 15
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 Peter Skands Parton Showers and NLO Matrix Elements 16
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 antenna functions) • 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 Peter Skands Parton Showers and NLO Matrix Elements 17
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 Peter Skands Parton Showers and NLO Matrix Elements 18
VINCIA Example: H ggg ► First Branching ~ first order in perturbation theory ► Unmatched shower varied from “soft” to “hard” : soft shower has “radiation hole”. Filled in by matching. Outlook: y 23 VINCIA 0. 008 Unmatched y 23 VINCIA 0. 008 Matched “soft” |A|2 radiation hole in high-p. T region Immediate Future: Paper about gluon shower Include quarks Z decays Matching y 23 VINCIA 0. 008 Unmatched “hard” |A|2 y 23 VINCIA 0. 008 Matched “hard” |A|2 Then: Initial State Radiation Hadron collider applications y 12 Peter Skands Parton Showers and NLO Matrix Elements y 12 19
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. T 1 >> p. T 2 >> p. T 3. . . • (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 Peter Skands Parton Showers and NLO Matrix Elements 20
Double Counting ► Combine different multiplicites inclusive sample? ► In practice – Combine X inclusive 1. [X]ME + showering 2. [X + 1 jet]ME + showering X+1 inclusive X+2 inclusive X exclusive ≠ X+1 exclusive X+2 inclusive 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+2 j, X+3 j, etc, this problem gets worse Peter Skands Parton Showers and NLO Matrix Elements 21
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. [Xn]ME + showering n inclusive 2. [Xn+1]ME + showering n+1 inclusive 3. … n+2 inclusive ► 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 njets • Check each parton from the Feynman diagram one jet? • If all partons are ‘matched’, keep event. Else discard it. n exclusive n+1 exclusive n+2 inclusive ► Virtue: can be done without knowledge of the internal workings of the generator. Only the fully showered final events are needed • Peter Skands Simple procedure to improve multijet rates in perturbative QCD Parton Showers and NLO Matrix Elements 22
- Slides: 22
