VINCIA

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VINCIA
Les Houches 2007

• Peter Skands
• Fermilab / Particle Physics Division /
  Theoretical Physics
• In collaboration with W. Giele, D. Kosower

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Aims

• Wed 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_at_NLO
  (NLO matching)
• We have done this by expanding on the ideas of
  Frixione, Nason, and Webber (MC_at_NLO), but with a
  few substantial generalizations

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Improved Parton Showers

• Step 1 A comprehensive look at the uncertainty
  (here PS _at_ 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 as)
• 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
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The Pure Shower Chain
Giele, Kosower, PS FERMILAB-PUB-07-160-T

• Shower-improved ( resummed) distribution of an
  observable
• Shower Operator, S (as a function of (invariant)
  time t1/Q)
• n-parton Sudakov
• Focus on dipole showers

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VINCIA
VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED
ANTENNAE
Giele, Kosower, PS FERMILAB-PUB-07-160-T
Gustafson, Phys. Lett. B175 (1986) 453

• 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?)

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Dipoles a dual description of QCD
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Lonnblad, Comput. Phys. Commun. 71 (1992) 15.
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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 ith dipole-antenna
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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 as unity)

pT-ordered Sudakov factor

• gg?ggg ?(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)
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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, thats 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
Initialization (PYTHIA 8 VINCIA) 1 event
Analytic, no splines 2s (lt 10-3s ?)
Analytic splines 2s lt 10-3s
Numeric, no splines 2s 6s
Numeric splines 50s lt 10-3s
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Matching

• X matched to n resolved partons at leading order
  and m lt 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 Xk
NLO matching term for Xk
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Matching to X1 at LO
Giele, Kosower, PS FERMILAB-PUB-07-160-T

• First order real radiation term from parton
  shower
• Matrix Element (X1 at LO above thad)
• ? Matching Term
• ? variations (or dead regions) in a2 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 MEs
• no dependence on unphysical cut or
  preclustering scheme (cf. CKKW)
• - not a complete order normalization changes (by
  integral of correction), but still LO

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Matching to X at NLO
Giele, Kosower, PS FERMILAB-PUB-07-160-T

• NLO virtual term from parton shower ( expanded
  Sudakov exp1 - )
• 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 a2
  not shower-specific
• Currently using Gehrmann-Glover (global) antenna
  functions
• Will include also Kosowers (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)

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Matching to X2 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)

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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
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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
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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

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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
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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 A2
VINCIA 0.008 Matched soft A2
y23
y23
radiation hole in high-pT region
y23
y23
VINCIA 0.008 Unmatched hard A2
VINCIA 0.008 Matched hard A2
y12
y12