parton showers and nlo matrix elements - …loopfest vi, fermilab, april 2007 parton showers and nlo...

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

Peter Skands Parton Showers and NLO Matrix Elements 2

OverviewOverviewParton Showers• QCD & Event Generators • Antenna Showers: VINCIA• Expansion of the VINCIA shower

Matching • LL shower + treelevel matching (through to s

2)• E.g. [X](0) , [X + jet](0) , [X + 2 jets](0) + shower (~ CKKW, but different)

• LL shower + 1loop matching (through to s)• E.g. [X](0,1) , [X + jet](0) + shower (~ MC@NLO, but different)

• A sketch of further developments


Main Tool• Approximate by truncation of perturbative series at fixed coupling

order

• Example:

QQuantumuantumCChromohromoDDynamicsynamics

Reality is more complicated


Traditional Event GeneratorsTraditional Event Generators

Basic aim: improve lowest order perturbation theory by including leading corrections exclusive event samples1. sequential resonance decays2. bremsstrahlung 3. underlying event4. hadronization5. hadron (and ) decays

E.g. PYTHIA2006: first publication of PYTHIA manual JHEP 0605:026,2006 (FERMILABPUB06052CDT)


TThe he BBottom ottom LLine ine

The S matrix is expressible as a series in gi, gin/tm, gi

n/xm, gin/mm, gi

n/fπm , …

To do precision physics:

Solve more of QCD Combine approximations which work in different regions: matching

Control itGood to have comprehensive understanding of uncertainties

Even better to have a way to systematically improve

Nonperturbative effects don’t care whether we know how to calculate them

FO DGLAP

BFKL

HQET

PT


Improved Parton ShowersImproved Parton ShowersStep 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 23 plane in CM)

• Vary the renormalization scheme (argument of 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


VINCIAVINCIA

VINCIA Dipole shower• C++ code for gluon showers

• Standalone since ~ half a year• Plugin to PYTHIA 8 (C++ PYTHIA) since ~ last week• Most results presented here use the plugin version

So far: • 2 different shower evolution variables:

• pTordering (~ ARIADNE, PYTHIA 8)

• Virtualityordering (~ 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” less wriggle room for nonpert 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 antennaeGiele, Kosower, PS : in progress


Checks: Checks: Analytic vs Numerical vs SplinesAnalytic vs Numerical vs SplinesCalculational methods1. 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 QR – coarse, fine, ultrafine)

Test example• Precision target: 106

• ggggg Sudakov factor (with nominal s = unity)

ggggg: (s,Q2)• Analytic• Splined

pTordered Sudakov factor

Numeric / AnalyticSpline (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 ~ 106

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)


Why Splines?Why Splines?Example: mH = 120 GeV

• Hgg + 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 handmade)• Aim to eventually handle any function & region numeric more general

< 10-3s50sNumeric + splines

< 10-3s2sAnalytic + splines

(< 10-3s ?)2sAnalytic, no splines

1 eventInitialization (PYTHIA 8 + VINCIA)

6s2sNumeric, no splines

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


MatchingMatchingMatching of up to one hard additional jet• PYTHIAstyle (reweight shower: ME = w*PS)

• HERWIGstyle (add separate events from ME: weight = MEPS)

• MC@NLOstyle (MEPS subtraction similar to HERWIG, but NLO)

Matching of generic (multijet) topologies (at tree level)• ALPGENstyle (MLM) • SHERPAstyle (CKKW)

• ARIADNEstyle (LönnbladCKKW)

• PATRIOTstyle (Mrenna & Richardson)

Brand new approaches (still in the oven)• Refinements of MC@NLO (Nason)

• CKKWstyle at NLO (Nagy, Soper)

• SCET approach (based on SCET – Bauer, Schwarz)

• VINCIA (based on QCD antennae – Giele, Kosower, PS)Evolution


MC@NLOMC@NLO

Nason’s approach:

Generate 1st 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)077MC@NLO in comparison

• Superior precision for total cross section• Equivalent to treelevel matching for event shapes (differences higher order)• Inferior to multijet matching for multijet topologies• So far has been using HERWIG parton shower complicated subtractions


Expanding the ShowerExpanding the ShowerStart from Sudakov factor

= Nobranching 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 spacesSums run over all possible branchings of all antennae


Matching at NLO: tree partMatching at NLO: tree partNLO real radiaton term from parton shower

Add extra treelevel 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 treelevel 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 spacesSums run over all possible branchings of all antennaeTwiddles = finite (subtracted) ME correctionsUntwiddled = divergent (unsubtracted) MEs


Matching at NLO: loop partMatching at NLO: loop partNLO 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 showerspecific• Currently using GehrmannGlover (global) antenna functions • Will include also Kosower’s (sector) antenna functions

Treelevel matching just corresponds to using zero

•(This time, too small |a| correction negative)


Matching at NNLO: tree partMatching at NNLO: tree partAdding more treelevel MEs is straightforward

Example: second emission term from NLO matched parton shower

X+2 jet treelevel ME correction term and matching equationMatching equation looks identical to 2 slides ago If all indices had been shown: subleading colour structures not derivable by nested 23 branchings do not get subtracted


Matching at NNLO: tree part, with 2Matching at NNLO: tree part, with 244Sketch only! • But from matching point of view at least, no problem to include 24

Second emission term from NLO matched parton shower with 24

• (For subleading colour structures, only |b|2 term enters)

Correction term and matching equation

• (Again, for subleading colour structures, only |b|2 term is nonzero)

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 24 consistently• If possible, would open the door to MC@NNLO


Under the RugUnder the RugThe simplified notation allowed to skip over a few issues we want to understand slightly better, many of them related• Start and restart 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


Under the Rug 2Under the Rug 2I 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 plugin for PYTHIA 8• We will then follow up with a writeup on the matching

I will just show an example based on treelevel matching for Hgg


VINCIA Example: H VINCIA Example: H gg gg ggg ggg

VINCIA 0.008Unmatched

“soft” |A|2

VINCIA 0.008Unmatched

“hard” |A|2

VINCIA 0.008Matched“soft” |A|2

VINCIA 0.008Matched

“hard” |A|2

y12

y23

y23

y23

y23

y12

First Branching ~ first order in perturbation theoryUnmatched shower varied from “soft” to “hard” : soft shower has “radiation hole”. Filled in by matching.

radiation hole in highpT region

Outlook:Immediate Future:

Paper about gluon showerInclude quarks Z decaysMatching

Then:Initial State RadiationHadron collider applications


A ProblemA ProblemThe 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 pT1 >> pT2 >> pT3 ...

• (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

wellseparated individual partons• Quickly becomes intractable beyond one loop and a handfull of legs• Unknown contributions from higher fixed orders


Double CountingDouble CountingCombine different multiplicites inclusive sample?

In practice – Combine1. [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 inclusiveX inclusiveX+1 inclusiveX+1 inclusive

X+2 inclusiveX+2 inclusive ≠X exclusiveX exclusiveX+1 exclusiveX+1 exclusiveX+2 inclusiveX+2 inclusive


The simplest example: ALPGENThe 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

2. [Xn+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 njets

• 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 inclusiven inclusiven+1 inclusiven+1 inclusive

n+2 inclusiven+2 inclusive

n exclusiven exclusiven+1 exclusiven+1 exclusiven+2 inclusiven+2 inclusive

parton showers and nlo matrix elements - …loopfest vi, fermilab, april 2007 parton showers and nlo...

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