loopfest vi, fermilab, april 2007 parton showers and nlo matrix elements peter skands fermilab /...

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

OverviewOverview► Parton Showers

• QCD & Event Generators

• Antenna Showers: VINCIA

• Expansion of the VINCIA shower

► Matching

• LL shower + tree-level matching (through to αs2)

• 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


► 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 decays

2. bremsstrahlung

3. underlying event

4. hadronization

5. hadron (and τ) decays

E.g. PYTHIA2006: first publication of PYTHIA manual JHEP 0605:026,2006 (FERMILAB-PUB-06-052-CD-T)


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 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

FO DGLAP

BFKL

HQET

χPT


Improved Parton ShowersImproved 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 23 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


VINCIAVINCIA

► VINCIA Dipole shower

• C++ code for gluon showers• Standalone since ~ half a year

• Plug-in to PYTHIA 8 (C++ PYTHIA) since ~ last week

• 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” less wriggle room for non-pert 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 antennae

Giele, Kosower, PS : in progress


Checks: Checks: Analytic vs Numerical vs SplinesAnalytic vs Numerical vs Splines

► Calculational 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: 10-6

• ggggg Sudakov factor (with nominal αs = unity)

ggggg: Δ(s,Q2)

• Analytic• Splined

pT-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 ~ 10-6

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 hand-made)

• Aim to eventually handle any function & region numeric more general

Initialization (PYTHIA 8 + VINCIA)

1 event

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

Analytic + splines 2s < 10-3s

Numeric, no splines 2s 6s

Numeric + splines 50s < 10-3s

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


MatchingMatching► 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)

• 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)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)077

► MC@NLO in comparison• 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


Expanding the ShowerExpanding the Shower► Start from Sudakov factor

= No-branching 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 spaces

Sums run over all possible branchings of all antennae


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

► Add extra tree-level 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 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

NB: simplified notation!

Differentials are over entire respective phase spaces

Sums run over all possible branchings of all antennae

Twiddles = finite (subtracted) ME corrections

Untwiddled = divergent (unsubtracted) MEs


Matching at NLO: loop partMatching at NLO: loop part► NLO 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 shower-specific• Currently using Gehrmann-Glover (global) antenna functions • Will include also Kosower’s (sector) antenna functions

Tree-level matching just corresponds to using zero

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


Matching at NNLO: tree partMatching 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 23 branchings do not get subtracted


Matching at NNLO: tree part, with 2Matching at NNLO: tree part, with 244

► Sketch 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 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 24 consistently

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


Under the RugUnder 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 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 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 Hgg


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

VINCIA 0.008

Unmatched

“soft” |A|2

VINCIA 0.008

Unmatched

“hard” |A|2

VINCIA 0.008

Matched

“soft” |A|2

VINCIA 0.008

Matched

“hard” |A|2

y12

y23

y23

y23

y23

y12

► 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-pT region

Outlook:

Immediate Future:

Paper about gluon shower

Include quarks Z decays

Matching

Then:

Initial State Radiation

Hadron collider applications


A ProblemA 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 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

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


Double CountingDouble Counting► Combine different multiplicites inclusive sample?

► In practice – Combine

1. [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 inclusive

X+1 inclusiveX+1 inclusive

X+2 inclusiveX+2 inclusive ≠X exclusiveX exclusive

X+1 exclusiveX+1 exclusive

X+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 inclusive

n+1 inclusiven+1 inclusive


n exclusiven exclusive

n+1 exclusiven+1 exclusive


loopfest vi, fermilab, april 2007 parton showers and nlo matrix elements peter skands fermilab /...

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