theory seminar, slac, may 2007 parton showers and nlo matrix elements peter skands fermilab /...

Theory Seminar, SLAC, May 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

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

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

• Leading-log 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 complicatedButch Cassidy and the Sundance Kid (1969) © 20th Century Fox Films


Monte Carlo Generators

Large-dimensional phase spaces

Monte Carlo integration:

Stratified sampling + stochastic error ~ N1/2 independent of dimension ‘events’

+ Markov Chain formulation of fragmentation:

1. Parton showers: iterative application of pertubatively calculable and universal kernels for n n+1 partons ( = resummation of soft/collinear Sudakov logarithms)

2. Hadronization: iteration of X X’ + hadron, phenomenological models based on known properties of nonperturbative QCD, lattice studies, and fits to data.

σX,inclMonte Carlo


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


► Starting observation: forward singularity of bremsstrahlung is universal Leading contributions to all radiation processes (QED & QCD)

can be worked out to all orders once and for all exponentiated (Altarelli-Parisi) integration kernels

► Iterative (Markov chain) formulation = parton shower• Generates the leading “collinear” parts of QED and QCD corrections to any

process, to infinite order in the coupling

• The chain is ordered in an “evolution variable”: e.g. parton virtuality, jet-jet angle, transverse momentum, …

a series of successive factorizations the lower end of which can be matched to a hadronization description at some fixed low hadronization scale ~ 1 GeV

Bremsstrahlung: Parton ShowersBremsstrahlung: Parton Showers

dσn+1 = dσn dΠnn+1 Pnn+1

dσn+2 = dσn (dΠnn+1 Pnn+1)2 and so on … exp[]

Schematic: Forward (collinear) factorization of QCD amplitudes exponentiation


Improved Parton ShowersImproved Parton Showers► Step 1: A comprehensive look at the uncertainty

• Vary the evolution variable (~ factorization scheme)

• Vary the radiation function (collinear limit only fixes the singular parts)

• Vary the kinematics map (explicit mapping from n to n+1 partons)

• Vary the renormalization scheme (argument of αs)

• Vary the infrared cutoff contour (hadronization cutoff)

► Step 2: Write a generator

• Make the above explicit (while still tractable) in a Markov Chain context parton shower MC algorithm


VINCIAVINCIA

► VINCIA Dipole shower

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

• Plug-in to PYTHIA 8 (C++ PYTHIA) since a few weeks

• (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 INTERACTING 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= no-branching probability,

generating function for shower

• 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


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 (PS)?

• 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 (ME)?• 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


Evolution

MatchingMatching► Matching of up to one hard additional jet, for specific processes

• PYTHIA-style (reweight shower: MEX+jet = w*PS)

• HERWIG-style (add separate X+jet events: w = MEX+jet-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 + “Quantum Monte Carlo” (Nagy, Soper)

• SCET approach (based on SCET – Bauer, Tackmann; Alwall, Mrenna, Schwarz)

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

ME: Matrix ElementPS: Parton Shower


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


Matched Parton ShowersMatched Parton Showers► Step 1: A comprehensive look at the uncertainty

• 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


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 radiation 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 antenna functions)

• 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► 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


Words of WarningWords of Warning► Still far from a complete description of hadron collisions

• We are really looking at just the first few terms in large expansions

• Non-perturbative physics not well understood

J. D. Bjorkenfrom a talk given at the 75th anniversary celebration of the Max-Planck Institute of Physics, Munich, Germany, December 10th, 1992. As quoted in: Beam Line, Winter 1992, Vol. 22, No. 4

[…] The Monte Carlo simulation has become the major means of visualization of not only detector performance but also of physics phenomena. So far so good. But it often happens that the physics simulations provided by the Monte Carlo generators carry the authority of data itself. They look like data and feel like data, and if one is not careful they are accepted as if they were data.

[…] They do allow one to look at, indeed visualize, the problems in new ways. But I also fear a kind of “terminal illness”, perhaps traceable to the influence of television at an early age. There the way one learns is simply to passively stare into a screen and wait for the truth to be delivered. A number of physicists nowadays seem to do just this.

theory seminar, slac, may 2007 parton showers and nlo matrix elements peter skands fermilab /...

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