Partial Wave Analysis WorkshopBeijing, Jan 25-26, 2007, IHEP

Klaus PetersGSI Darmstadt & University Frankfurt

- Potential Problems and Instabilities in Amplitude Analyses- Significance and Quality Assurance of Spin-Parity Analyses

Significance and Quality Assurance of Spin-Parity Analyses

Phase space representationFit methods and strategies

Background treatmentRecommendations

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Not a cut and count experiment...

If you do have a counting experiment the strategy is clearIf you have a theoretical expectation things become even easierIn our case we don‘t have this

Ideallywe have a description which we verify or falsify

if we would have this, hadron physics is solved and experiments are obsolete

We needeither a theoretically well defined Ansatzor a almost model independent description of the data

should still reveal the evident physics from the data

This requiresa critical review on what we are doing and in how far we can trust the results Systematics, Quality, Significance

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But life isn‘t that easy

Ansatz very often driven by technical feasibility rather then correct physicsthe correct approach might be to complicatedor the measurement is not complete

missing observablesmissing channels in coupled channel approachesin those cases it is not simple to accommodate this correctly

In a general approach Singularities appearhave to be cancelled outit’s sometimes not easy because they are not always obviousneed for algebraic construction of probabilities

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This is not an exhaustive list...

but a collection of observationswhich lead to a few things people should obeyand a general strategy for developments

topics for this talkphase space considerations – binningmass variationsfit method/functionbackground treatmentfit strategiesminimization kernelbranching fractionsquality estimators (significance)recommended PWA system

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Phase space considerations

If no. of observablesd is limited and statistics is high binned data

Binning is simple...is this true? no

WhyPhase space distributions from experiments are not flatSymmetries in the final states have to be taken care ofBinning has to match the resolution

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Non-Flat distributions: Adaptive binning

cut-off

Finite size effects in a bin of the Dalitz plotlimited line shape sensitivity for narrow resonances

Entry cut-off for bins of a Dalitz plotsχ2 makes no sense for small #entriescut-off usually 10 entries

Problemsthe cut-off method may deplete important regions of the plot to muchcircumvent this by using a bin-by-binPoisson-test for these areas

alternatively: adaptive Dalitz plots, but one may miss narrow depleted regions, like the f0(980) dipsystematic choice-of-binning-errors

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Boundary problem I

Most Dalitz plots are symmetric (2dim: 1-fold, 2-fold or 6-fold)Problem: sharing of events

Solution: transform DP

r

f(r)

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Boundary Problem II

Efficiencies often factorize in mass and angular distribution

2nd ApproachUse mass and cosθNot always applicable(if symmetries are involved)

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Finite Resolution...

Due to resolution or wrong matchingTrue phase space coordinates of MC events are different from the reconstructed coordinatesIn principle amplitudes of MC-events have to be calculated at the generated coordinate, not the reconstructed locationBut they are plotted at the reconstructed location

Applies toExperiments with “bad” resolution (like Asterix)For narrow resonances [like Φ or f1(1285) or f0(980)]

Wrongly matched tracksbut also to high resolution experiments with large tails for a small amount of the events

For binned data it cures phase-smearing and non-isotropic resolution effects

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Finite Resolution in Likelihood fits

Resonances always appear as a convolution of resolution and resonance properties

PWA model has to introduce resolution if neccessarythus a pure BWigner or sharp resonances in K-matrices have an inherent problem

since amplitude is always calculated at reconstructed phase space location

Formally in that caseone has to do a MC intergration of the probability according to resolutionsmearing

at least 10 points dramatic increase in CPU timeactual # of points depend on ratio resolution to FWHM (feature

size)

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How to treat “well known“ resonances?

There are two approaches to thattake it from the PDG and your done

but your experiment may demand a systematic shift bias to the likelihood function

if this is a detector effect you may find a reason, but sometimes the systematic studies are an uneccessary complication

other approach, take nothing for granted fit everything at the end

As always:The truth is probably in between

one approach might be to include a probability function in the likelihood function to accommodate a marginal change in resonance properties and let it vary freely

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Fit methods - χ2 vs. Likelihood

χ2

small # of independent phase space observables usually not more than 2

High statistics >10k if there are only a few well known resonances>50k for complicated final states with discovery potentiale.g. CB found 752.000 events of the type pp3π0

Significance is easy to calculate

-logLmore than 2 independent phase space observableslow statistics (compared to size of phase space)narrow structures [like Φ(1020)] if adaptive binning does not work

Systematic check use both if applicable

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Background subtraction and/or fitting

Experiments at LEAR did a great job, but backgrounds were low and statistics were extremely high

Background was usually not an issueIn D(s)-Decays we know this is a severe problem

Backgrounds can exceed 50%

ApproachesLikelihood compensation

add logLi of all background events (from sidebands)

not so simple, convergence is unclear

Background parameterization (added incoherently)combined fitfit to sidebands and fix for Dalitz plot fit

Try all to get a feeling on the systematic error

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Strategy

Where to start the fit

Is one more resonance significant

Indications for a bad solution

Where to stop the sophistication/fit

?

??

?

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Where to start

Problem dependentstart with obvious signatures

Sometimes a moment-analysis can help to find important contributionsbest suited if no crossing bands occur

In the future this might be possible with genetic algorithms in all casesParametrize wave bin by bin and perform a genetic search for a reasonable desciption of the data (?500 params)

( ) ( )

( ) ( )

LM0

LM0

t LM D φ,θ,0

I Ω D φ,θ,0 dΩ

=

= òD0KSK+K-

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Where to stop

Apart from what was said beforeAdditional hypothetical trees (resonances, mechanisms) do not improve the description considerablyDon‘t try to parameterize your data with inconsistent techniquesIf the model don‘t match, the model might be the problemreiterate with a better model

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Indications for a bad solution

Apart from what was said beforeone indication can be a large branching fraction of interference terms

Definition of BF of channel j BFj = ∫|Aj|2dΩ/∫|ΣiAi|2

But due to interferences, something is missingIncoherentI=|A|2+|B|2

CoherentI=|A+eiφB|2 = |A|2+|B|2+2[Re(AB*)sinφ+Im(AB*)cosφ

If ΣjBFj is much different from 100% there might be a problemThe sum of interference terms must

vanish if integrated from -∞ to +∞But phase space limits this regionIf the resonances are almost covered by phase spacethen the argument holds......and large residual interference intensities signal overfitting

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Is one more resonance significant ?

Base your decision onobjective bin-by-bin χ2 and χ2/Ndof

visual qualityis the trend right?is there an imbalance between different regions

compatibility with expected L structureIs the spin-paritiy of well-known resonances unambiguous?

Produce Toy MC for Likelihood Evaluationmany sets with full efficiency and Dalitz plot fit

each set of events with various amplitude hypothesescalc L expectation

L expectation is usually not just ½/dofsometimes adding a wrong (not necessary) resonance

can lead to values over 100!compare this with data

Result: a probability for your hypothesis!# of fits giving back the truth tell you the confidence level

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ToyMC Significance Test

Your experiment may yield a certain likelihood patternHypo 1–log1=-5123

Hypo 2–log2=-4987 (ΔL=136)

Hypo 3–log3=-4877 (ΔL=110)

Is Hypo 3 really needed? What is the significanceToyMC create independent toy data sets which have exactly the same composition as solutions 1,2 and 3If 3 is the right solution find out how often –log3 is smaller than –log2, the percentage gives the confidence level significance

table from PDG06 for ±δ

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Branching fractions

For a single wave from a single initial statelike all 0++ or all 1–- one may use the formula

BFj = ∫|Aj|2dΩ/∫|ΣiAi|2

Σi BFj ≠ 100 %

Alternatively

BFj = ∫|Aj|2(1+Σj2[Re(AiAj*)sinφij+Im(AiAj

*)cosφij)dΩ/∫|ΣiAi|2

Σi BFj = 100 %

For a single resonance in a wave things are much more complicated

isolated resonances use individual amplitudes and use definition abovecoupled channels no well defined proceduredepends strongly if there is a clear relationship between couplings and T-Matrix singularities

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Proposed PWA system

GoalsFlexible

should serve all hadron spectroscopy communities

Fast there is no time to waste

Platform independent possible by design, framework in root

Easy to use root C++-Scripting

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Flexibility

A static approach is the wrong solutionmost implementations are experiment and/or reaction specificlimited use outside of a collaborationdebug level can be improved if widely used

If we can get such a system it is much easier to validate the results of one experiment

and it is much easier to combine experiments/channelsthis needs a mechanism to combine fits

We also have to accommodatevarious production modesvarious angular formalisms

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Speed

Whoever has done a PWA or Dalitzplot fit knows by heart

SPEED is an ISSUEWe face in BESIII, GlueX and PANDA extremely high statisticsDue to the many observables in many cases a binned fit is impossible and the likelihood fit easily occupies all resources you have

you have to be cleverTypical techniques

control the minimization strategystop intermediate steps early

minimize calculationscalculate only things which have changed since the last

calculation

use algebraic derivativesutilize multiple compute nodes

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Platform independence/Usability

Well defined interfaces for theangular functionsdynamical functionsparameter exchange from functions to minimizer and v.v.automated documentation of the fitfit results in text and plotsinternal toolkits

T-pole finderT-pole mover (Riemann sheet analysis)Dalitzplot transformer

For a real implementation one needsfunction constructor setusing this set

implementations for common angular formalismslibrary of dynamical functions (from literature and new

developments)

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Function constructor set

All functions represent terms in a formulae.g. (1+x)y2

they are objectsand are derived from a basic function objectthey knows about their mothers and daughters

e.g. function (1+x)y2 can be decomposed intof1=1+x knows daughter function x, knows mother f3

f2=y2 knows daughter function y, knows mother f3

f3=f1 f2 knows daughter function f1;f2, knows mother

each node saves its intermediate results (persistency)special functions needed to use phasespace imformation as well as fitable and non-fitable parameters

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Minimization Kernel

The usual choice is MINUIT, but there are several reasons for a change

more access to the actual minimization process and the progress is neededuse of algebraic (1st and 2nd) derivatives should be possibleparameter objects need to know if they have changedparameters ned to be more abstract (also integer and boolean)fits need persistency mechanisms (save and retrieve)

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A Networking Approach

Euorpean Commission Framework Programme 7 (FP7) Joint Research Activities (JRA)

under the umbrella of an Integrated Infrastructure Initiative (III, I3)

Ours is I3HP (Chair: C. Guaraldo, LNF, Frascati, Italy) under FP6 Started collecting “Expression of Interest“ in new activities

Ideal framework to start a collaboration on PWA programmes and techniques New programme to start in 2008 (for 4-6 years) Funds personnel, travel and running costs Own contribution now only 25% (50% in FP6)

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Expression of Interest

Draft 0.1 Darmstadt, November 30, 2006 Coordinator Klaus Peters (GSI)

Proposal for a Joint Research Activity of the 7th Framework

“Amplitude analysis for high precision hadron spectroscopy”Subjects:

Unified coupled channel analysis methods including analyticity, unitarity, crossing symmetry and gauge invarianceTool packages for comprehensive and experiment independent amplitude analysisFitting and validation algorithms and procedures to ensure high quality results from forthcoming high statistics experiments

Interested institutions:Bonn University (HISKP); Cracow University; University Durham; GSI Darmstadt (Hadron physics experiment and theory); Pavia University; Technical University Munich (E18); KVI Groningen (Theory)

Interested International Partners (non-EU):IHEP Beijing, China; Indiana University, Bloomington, U.S.A.

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Deliverables and Requested Funding

Deliverables:A “Yellow Book” on amplitude analysis with

a critical review of developments and applications in the past 40 yearsnew theoretical and phenomenological trends and developmentsapplication to high precision and high statistics data

Various Software Librariescore system for dynamical functions (with standardized interfaces)library of dynamical functionshigh performance amplitude construction framework

Requested funding:Personnel

typically equivalent 1 postdoc plus 1-2 PhD students per group)Travel

Support for 3 international workshops (2008/9/10) organized by this JRATravel expenses for the meetings of the JRA (once a year)Per-diem and travel expenses for guests

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Many Thanks

Tothe audience

and the organizers