les houches 2007 vincia peter skands fermilab / particle physics division / theoretical physics in...

Les Houches 2007

VINCIA

Peter Skands

Fermilab / Particle Physics Division / Theoretical Physics

In collaboration with W. Giele, D. Kosower

Peter Skands Parton Showers and NLO Matrix Elements 2

AimsAims► 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 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


Improved Parton ShowersImproved 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 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


The Pure Shower ChainThe 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 showersDipole branching phase space

“X + nothing” “X+something”

Giele, Kosower, PS : FERMILAB-PUB-07-160-T


VINCIAVINCIA

► 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?)

Dipoles – a dual description of QCD

1

3

2

VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE

Giele, Kosower, PS : FERMILAB-PUB-07-160-TGustafson, Phys. Lett. B175 (1986) 453

Lonnblad, Comput. Phys. Commun. 71 (1992) 15.


Dipole-Antenna FunctionsDipole-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

yar = sar / si

si = invariant mass of i’th dipole-antenna


Gehrmann-De Ridder, Gehrmann, Glover, JHEP 09 (2005) 056


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 not 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 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 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► “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



Matching to X+1 at LOMatching to X+1 at LO► 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



Matching to X at NLOMatching 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)



?

Matching to X+2 at LOMatching 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 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 23 branchings do not get subtracted



Going deeper?Going deeper?

► NLL Sudakov with 24

• 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



Universal HadronizationUniversal 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 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



““Sudakov” vs LUCLUS pSudakov” vs LUCLUS pTTGiele, Kosower, PS : FERMILAB-PUB-07-160-T

Vincia “hard” & “soft”

Vincia nominal

Pythia8

Same variations

2-jet rate vs PYCLUS pT (= LUCLUS ~ JADE)

Preliminary!


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

Automated matching

Then:

Initial State Radiation

Hadron collider applications


les houches 2007 vincia peter skands fermilab / particle physics division / theoretical physics in...

Documents

nlo nlo

nlo matrix elements

peter skandsparton showers

dipole showers dipole

kosower slide

t slide

pure shower chain shower

shower operator