A Primer on Partial Wave Analysis in hadron spectroscopy Charm 2006 International Conference on Tau-Charm Physics Beijing, June 5-7 Klaus Peters IKF, JWGU.

A Primer on Partial Wave Analysisin hadron spectroscopy

Charm 2006International Conference on Tau-Charm Physics

Beijing, June 5-7Klaus PetersIKF, JWGU Frankfurt und KP3, GSI Darmstadt

2

Klaus Peters - PWA Primer

Overview

What do we need to talk about ?

Introduction and Concepts

Spin Formalisms

Dynamical Functions

Technical Issues

3


Overview – Introduction and Concepts



Spin Formalisms

Dynamical Functions

Technical Issues

GoalsWave ApproachIsobar-ModelLevel of Detail

4


Overview – Spin Formalisms



Spin Formalisms

Dynamical Functions

Technical Issues

Overview

Zemach Formalism

Canonical Formalism

Helicity Formalism

Moments Analysis

5


Overview - Dynamical Functions



Spin Formalisms

Dynamical Functions

Technical Issues

Breit-Wigner

S-/T-Matrix

K-Matrix

P/Q-Vector

N/D-Method

Barrier Factors

Interpretation

6


Overview – Technical Issues / Fitting



Spin Formalisms

Dynamical Functions

Technical Issues

Coding Amplitudes

Speed is an Issue

Fitting Methods

Caveats

FAQ

7


(non-theorist) References

• hep-ex/0410014: 18p., D. Asner, Charm Dalitz Plot Analysis Formalism and ResultsExpanded version of review in "Review of Particle Physics", S. Eidelman et al., Phys. Lett. B 592, 1 (2004)

• hep-ph/0412069: 62p., K. Peters, A Primer on Partial Wave AnalysisLectures given at International Enrico Fermi School of Physics, Varenna, Italy, 6-16 Jul 2004. published Varenna 2004, Hadron physics p. 451-514

• Charm 2006D. AsnerM. PappagalloM. Pennington

8


Overview – Technical Issues / Fitting



Spin Formalisms

Dynamical Functions

Technical Issues

9


What is the mission ?

• Particle physics at small distances is well understoodOne Boson Exchange, Heavy Quark Limits

• This is not true at large distancesHadronization, Light mesonsare barely understood compared to their abundance

• Understanding interaction/dynamics of light hadrons willimprove our knowledge about non-perturbative QCDparameterizations will give provide toolkit to analyze heavy quark processesthus an important tool also for precise standard model tests

• We needAppropriate parameterizations for the multi-particle phase spaceA translation from the parameterizations to effective degrees of freedom for a deeper understanding of QCD

10


Goal

• For whatever you need the parameterization of the n-Particle phase spaceIt contains the static properties of the unstable (resonant) particles within the decay chain likemasswidthspin and parities

as well as properties of the initial stateand some constraints from the experimental setup/measurement

• The main problem is, you don‘t need just a good description,you need the right oneMany solutions may look alike but only one is right and they differ strongly in the phases involved

11


Intermediate State Mixing

• Many states may contribute to a final statenot only ones with well defined (already measured) propertiesnot only expected ones

• Many mixing parameters are poorly knownK-phasesSU(3) phases

• In additionalso D/S mixing(b1, a1 decays)

12


n-Particle Phase space, n=3

• 2 ObservablesFrom four vectors12Conservation laws -4Meson masses -3Free rotation -3Σ 2

• Usual choiceInvariant mass m12

Invariant mass m13

π3

π2pp

π1

Dalitz plot

13


J/ψ π+π-π0

• Angular distributions are easily seen in the Dalitz plot

cosθ

-1 0 +1

14


It’s All a Question of Statistics ...

• pp 30 with

• 100 events

15


It’s All a Question of Statistics ... ...

• pp 30 with

• 100 events

• 1000 events

16


It’s All a Question of Statistics ... ... ...

• pp 30 with

• 100 events

• 1000 events

• 10000 events

17


It’s All a Question of Statistics ... ... ... ...

• pp 30 with

• 100 events

• 1000 events

• 10000 events

• 100000 events

18


Introducing Partial Waves

• Schrödinger‘s Equation

Angular Amplitude

Dynamic Amplitude

19


Spin Formalisms – on overview

• Tensor formalismsin non-relativistic (Zemach) or covariant formFast computation, simple for small L and S

• Spin-projection formalismswhere a quantization axis is chosen and proper rotations are used to define a two-body decayEfficient formalisms, even large L and S easy to handle

• Formalisms based on Lorentz invariants (Rarita-Schwinger)where each operator is constructed from Mandelstam variables onlyElegant, but extremely difficult for large L and S

20


Argand Plot

21


Standard Breit-Wigner

• Full circle in the Argand Plot

• Phase motion from 0 to π

Intensity I=ΨΨ*

Phase δ Speed dφ/dm

Argand Plot

22


Breit-Wigner in the Real World

• e+e- ππ

mππ

ρ-ω

23


Dynamical Functions are Complicated

• Search for resonance enhancements is a major tool in meson spectroscopy

• The Breit-Wigner Formula was derived for a single resonance appearing in a single channel

• But: Nature is more complicatedResonances decay into several channelsSeveral resonances appear within the same channelThresholds distort line shapes due to available phase space

• A more general approach is needed for a detailed understanding

24


Isobar Model

• Generalizationconstruct any many-body system as a tree of subsequent two-body decaysthe overall process is dominated by two-body processesthe two-body systems behave identical in each reactiondifferent initial states may interfere

• We needneed two-body “spin”-algebravarious formalisms

need two-body scattering formalismfinal state interaction, e.g. Breit-Wigner

Isobar

25


K-Matrix Definition

• T is n x n matrix representing n incoming and n outgoing channel

• If the matrix K is a real and symmetric

• also n x n

• then the T is unitary by construction

26


Example: 1x2 K-Matrix Nearby Poles

• Two nearby poles (1.27 and 1.5 GeV/c2)

• show nicely the effect of unitarization

2 BWK-Matrix

27


Flatté

• Example

• a0(980) decaying into πη and KK

BW πηFlatte πηFlatte KK

Intensity I=ΨΨ* Phase δ

Real PartArgand Plot

28


Example: K-Matrix Parametrizations

• Au, Morgan and Pennington (1987)

• Amsler et al. (1995)

• Anisovich and Sarantsev (2003)

29


Experimental Techniques

Scattering Experiments

• πN - N* measurement

• πN - meson spectroscopyE818, E852 @ AGS, GAMS

• pp meson threshold productionWasa @ Celsius, COSY

• pp or πp in the central regionWA76, WA91, WA102

• γN – photo productionCebaf, Mami, Elsa, Graal

“At-rest” Experiments

• pN @ rest at LEARAsterix, Obelix, Crystal Barrel

• J/ψ decaysMarkIII,DM2,BES,CLEO-c

• ф(1020) decaysKloe @ Dafne, VEPP

• D and Ds decays

FNAL, Babar, Belle

30




• partial waves decomposition via moment analysis

• systematic studies to limit#waves

• dynamics appear as amplitude variations

• resonance parameters from fits to amplitudes


• ad-hoc introduction of waves

• ad-hoc introduction of dynamic amplitudes (“resonances”)

• systematic studies to limit #waves and #resonances

• resonance parameters appear as fit parameters

31




• exchange model needed

• ad-hoc intermediate resonances parameters fixed for wave decomposition


• independent of production model

• intermediate resonances treated identically to final state resonances

• crossing bands may provide high resolution interferometer

32


Moments Analysis

• Consider reaction

• Total differential cross section

• expand H

• leading to

33


E852 f1π and b1π

blue one 1-+ poleblack two poles2=70.6/47=1.5

E852

E852

E852

34


Fit for D0Ksπ+π-

see M. Pappagallo, this conf.

35


• Overlapping band usually make it very difficult to do a moment analysis to get an impression on the wave content

36


Momentum Analysis in a Dalitz Plot

• In some cases it‘s possible if no sharp bands overlap

0 2 20

01 SP

0 22

4 Y S P

4 Y 2 S P cos

24 Y P

5

π

π φ

π

see M. Pappagallo, this conf.

37


Technical Aspects

• Initial composition for fits ?

• How to treat (non-peaking) background ?

• How to identifiy the best fit ?

• Numerical aspects for complex dynamical functions ?

• Speed Issues, Precision of MC phase space integrals !

• Systematic aspects of scalar waves !

38


Summary

• Partial Wave decompostion and proper treatment of resonances (dynamics, spins, parities) has become extremely important in charm and beauty physics

• Required for a proper understanding multibody D(S) decays

• In particularMeasurement of γ/Φ3 via DKSππ as interferometer

charm-Mixing from time-dependent Dalitzplot fits

• For this purpose: one needs a correct parameterization which yield correct phases !

2m

2m

2m

2m 0D 0D

2 0 2Sm m(K )π-

- = 2 0 2Sm m(K )π+

+ =

A Primer on Partial Wave Analysis in hadron spectroscopy Charm 2006 International Conference on Tau-Charm Physics Beijing, June 5-7 Klaus Peters IKF, JWGU.

Documents

klaus peters ikf

review of particle physics

hadron physics

nparticle phase spaceit

partial wave analysislectures

heavy quark limitsthis

heavy quark processesthus

taucharm physicsbeijing