Transcript Standard Model Higgs - University at Buffalo

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
Overview
► 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
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Parton Showers and NLO Matrix Elements
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Q
uantum
C
hromo
D
ynamics
► Main Tool
• Approximate by truncation of perturbative series at fixed coupling
order
• Example:
Reality is more complicated
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Traditional Event Generators
► Basic aim: improve lowest order perturbation theory by
including leading corrections  exclusive event samples
1. sequential resonance decays
2.
3.
4.
5.
Peter Skands
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)
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T B
he
ottom
L
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
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Parton Showers and NLO Matrix Elements
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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
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VINCIA
virtual numerical collider with interesting 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:
• pT-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 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?)
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Checks: Analytic vs Numerical vs Splines
► Calculational methods
1. Analytic integration over
resolved region (as defined
by evolution variable) –
pT-ordered Sudakov factor
ggggg: Δ(s,Q2)
• 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
(Pythia8 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 (3x1000 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  no
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)
Initialization
Have to do it only once
for each spline point
during initialization
1 event
(PYTHIA 8 + VINCIA)
Analytic, no splines
2s
(< 10-3s ?)
Analytic + splines
2s
< 10-3s
Numeric, no splines
2s
6s
Numeric + splines
50s
< 10-3s
•
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
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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
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MC@NLO
Frixione, Nason, Webber, JHEP 0206(2002)029 and 0308(2003)007
Nason’s
approach:
JHEP 0411(2004)040
Generate 1st 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
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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):
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Matching at NLO: tree part
► NLO real radiaton 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
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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
•
•
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Currently using Gehrmann-Glover (global) antenna functions
Will include also Kosower’s (sector) antenna functions
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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
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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
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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 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
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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
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VINCIA Example: H  gg  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:
y23
VINCIA 0.008
Unmatched
y23
VINCIA 0.008
Matched
“soft” |A|2
“soft” |A|2
radiation
hole in
high-pT
region
Immediate Future:
Paper about
gluon shower
Include quarks
 Z decays
Matching
y23
VINCIA 0.008
Unmatched
“hard” |A|2
y23
VINCIA 0.008
Matched
“hard” |A|2
Then:
Initial State
Radiation
Hadron collider
applications
y12
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y12
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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 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
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Double Counting
► Combine different multiplicites  inclusive sample?
► In practice – Combine
X inclusive
1. [X]ME + showering
X+1 inclusive
2. [X + 1 jet]ME + showering
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+2j, X+3j, etc, this problem gets worse
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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
n exclusive
•
Check each parton from the Feynman diagram  one jet?
n+1 exclusive
•
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
•
Peter Skands
n+2 inclusive
 Simple procedure to improve multijet rates in perturbative QCD
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