Transcript A Primer for Partial Wave Analysis in hadron spectroscopy

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 Frankfurt und KP3, GSI Darmstadt
Overview
Klaus Peters - PWA Primer
What do we need to talk about ?
Introduction and Concepts
Spin Formalisms
Dynamical Functions
Technical Issues
2
Overview – Introduction and Concepts
Klaus Peters - PWA Primer
What do we need to talk about ?
Goals
Wave Approach
Isobar-Model
Level of Detail
Introduction and Concepts
Spin Formalisms
Dynamical Functions
Technical Issues
3
Overview – Spin Formalisms
Klaus Peters - PWA Primer
What do we need to talk about ?
Overview
Zemach Formalism
Canonical Formalism
Helicity Formalism
Moments Analysis
4
Introduction and Concepts
Spin Formalisms
Dynamical Functions
Technical Issues
Overview - Dynamical Functions
Klaus Peters - PWA Primer
What do we need to talk about ?
Breit-Wigner
S-/T-Matrix
K-Matrix
P/Q-Vector
N/D-Method
Barrier Factors
Interpretation
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Introduction and Concepts
Spin Formalisms
Dynamical Functions
Technical Issues
Overview – Technical Issues / Fitting
Klaus Peters - PWA Primer
What do we need to talk about ?
Introduction and Concepts
Spin Formalisms
Coding Amplitudes
Speed is an Issue
Fitting Methods
Caveats
FAQ
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Dynamical Functions
Technical Issues
(non-theorist) References
Klaus Peters - PWA Primer
• hep-ex/0410014: 18p., D. Asner, Charm Dalitz Plot Analysis
Formalism and Results
Expanded 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
Analysis
Lectures given at International Enrico Fermi School of Physics,
Varenna, Italy, 6-16 Jul 2004.
published  Varenna 2004, Hadron physics p. 451-514
• Charm 2006
D. Asner
M. Pappagallo
M. Pennington
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Overview – Technical Issues / Fitting
Klaus Peters - PWA Primer
What do we need to talk about ?
Introduction and Concepts
Spin Formalisms
Dynamical Functions
Technical Issues
8
What is the mission ?
Klaus Peters - PWA Primer
• Particle physics at small distances is well understood
One Boson Exchange, Heavy Quark Limits
• This is not true at large distances
Hadronization, Light mesons
are barely understood compared to their abundance
• Understanding interaction/dynamics of light hadrons will
improve our knowledge about non-perturbative QCD
parameterizations will give provide
toolkit to analyze heavy quark processes
thus an important tool also for precise standard model tests
• We need
Appropriate parameterizations for the multi-particle phase space
A translation from the parameterizations to effective degrees of
freedom for a deeper understanding of QCD
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Goal
Klaus Peters - PWA Primer
• For whatever you need the parameterization
of the n-Particle phase space
It contains the static properties of the unstable (resonant) particles
within the decay chain like
mass
width
spin and parities
as well as properties of the initial state
and some constraints from the experimental setup/measurement
• The main problem is, you don‘t need just a good description,
you need the right one
Many solutions may look alike but only one is right
and they differ strongly in the phases involved
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Intermediate State Mixing
Klaus Peters - PWA Primer
• Many states may
contribute to a final state
not only ones with
well defined (already
measured) properties
not only expected ones
• Many mixing parameters
are poorly known
K-phases
SU(3) phases
• In addition
also D/S mixing
(b1, a1 decays)
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n-Particle Phase space, n=3
Klaus Peters - PWA Primer
• 2 Observables
From four vectors
Conservation laws
Meson masses
Free rotation
Σ
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-4
-3
-3
2
Dalitz plot
• Usual choice
Invariant mass m12
Invariant mass m13
π1
pp
π2
π3
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J/ψ  π+π-π0
Klaus Peters - PWA Primer
• Angular distributions are easily seen in the Dalitz plot
cosθ
-1
13
0
+1
It’s All a Question of Statistics ...
Klaus Peters - PWA Primer
• pp  3p0 with
• 100 events
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It’s All a Question of Statistics ... ...
Klaus Peters - PWA Primer
• pp  3p0 with
• 100 events
• 1000 events
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It’s All a Question of Statistics ... ... ...
Klaus Peters - PWA Primer
• pp  3p0 with
• 100 events
• 1000 events
• 10000 events
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It’s All a Question of Statistics ... ... ... ...
Klaus Peters - PWA Primer
• pp  3p0 with
• 100 events
• 1000 events
• 10000 events
• 100000 events
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Introducing Partial Waves
Klaus Peters - PWA Primer
• Schrödinger‘s Equation
Angular Amplitude
Dynamic Amplitude
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Spin Formalisms – on overview
Klaus Peters - PWA Primer
• Tensor formalisms
in non-relativistic (Zemach) or covariant form
Fast computation, simple for small L and S
• Spin-projection formalisms
where a quantization axis is chosen and proper rotations are used to
define a two-body decay
Efficient formalisms, even large L and S easy to handle
• Formalisms based on Lorentz invariants (Rarita-Schwinger)
where each operator is constructed from
Mandelstam variables only
Elegant, but extremely difficult for large L and S
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Argand Plot
Klaus Peters - PWA Primer
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Standard Breit-Wigner
Klaus Peters - PWA Primer
• Full circle in the
Argand Plot
Intensity I=ΨΨ*
Phase δ
Speed dφ/dm
Argand Plot
• Phase motion
from 0 to π
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Breit-Wigner in the Real World
Klaus Peters - PWA Primer
• e+e- ππ
mππ
ρ-ω
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Dynamical Functions are Complicated
Klaus Peters - PWA Primer
• 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 complicated
Resonances decay into several channels
Several resonances appear within the same channel
Thresholds distort line shapes due to available phase space
• A more general approach is needed
for a detailed understanding
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Isobar Model
Klaus Peters - PWA Primer
• Generalization
construct any many-body system
as a tree of subsequent two-body decays
the overall process is dominated
by two-body processes
the two-body systems behave
identical in each reaction
different initial states may interfere
• We need
need two-body “spin”-algebra
various formalisms
need two-body scattering formalism
final state interaction, e.g. Breit-Wigner
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Isobar
K-Matrix Definition
Klaus Peters - PWA Primer
• 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
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Example: 1x2 K-Matrix Nearby Poles
Klaus Peters - PWA Primer
2 BW
K-Matrix
• Two nearby poles (1.27 and 1.5 GeV/c2)
• show nicely the effect of unitarization
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Flatté
Klaus Peters - PWA Primer
Argand Plot
Real Part
BW πη
Flatte πη
Flatte KK
• Example
• a0(980) decaying
into πη and KK
Intensity I=ΨΨ*
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Phase δ
Example: K-Matrix Parametrizations
Klaus Peters - PWA Primer
• Au, Morgan and Pennington (1987)
• Amsler et al. (1995)
• Anisovich and Sarantsev (2003)
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Experimental Techniques
Klaus Peters - PWA Primer
Scattering Experiments
• πN - N* measurement
• πN - meson spectroscopy
E818, E852 @ AGS, GAMS
• pp meson threshold production
Wasa @ Celsius, COSY
• pp or πp in the central region
WA76, WA91, WA102
• γN – photo production
Cebaf, Mami, Elsa, Graal
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“At-rest” Experiments
• pN @ rest at LEAR
Asterix, Obelix, Crystal Barrel
• J/ψ decays
MarkIII,DM2,BES,CLEO-c
• ф(1020) decays
Kloe @ Dafne, VEPP
• D and Ds decays
FNAL, Babar, Belle
Experimental Techniques
Klaus Peters - PWA Primer
Scattering Experiments
“At-rest” Experiments
• partial waves decomposition
• ad-hoc introduction of waves
• systematic studies to limit
• ad-hoc introduction of dynamic
• dynamics appear as amplitude
• systematic studies to limit
• resonance parameters from fits
• resonance parameters appear as
 via moment analysis
#waves
variations
to amplitudes
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amplitudes (“resonances”)
#waves and #resonances
fit parameters
Experimental Techniques
Klaus Peters - PWA Primer
Scattering Experiments
“At-rest” Experiments
• exchange model needed
• independent of production model
• ad-hoc intermediate resonances • intermediate resonances treated
 parameters fixed for wave
decomposition
 identically to final state
resonances
• crossing bands may provide high
resolution interferometer
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Moments Analysis
• Consider reaction
• Total differential cross section
• expand H
• leading to
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Klaus Peters - PWA Primer
E852 f1π and b1π
Klaus Peters - PWA Primer
E852
E852
E852
blue one 1-+ pole
black two poles
c2=70.6/47=1.5
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Fit for D0Ksπ+πsee M. Pappagallo, this conf.
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Klaus Peters - PWA Primer
Klaus Peters - PWA Primer
• Overlapping band usually make it very difficult to do a moment
analysis to get an impression on the wave content
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Momentum Analysis in a Dalitz Plot
see M. Pappagallo, this conf.
Klaus Peters - PWA Primer

 4π Y00  S 2  P 2


0
 4π Y1  2 S P cos φ SP

 4π Y 0  2 P 2
2

5
• In some cases it‘s possible if no
sharp bands overlap
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Technical Aspects
Klaus Peters - PWA Primer
•
•
•
•
•
•
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 !
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Summary
Klaus Peters - PWA Primer
• 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 particular
Measurement of γ/Φ3 via DKSππ as interferometer
charm-Mixing from time-dependent Dalitzplot fits
• For this purpose: one needs a correct parameterization which
yield correct phases !
m2- = m(K0Sπ- )2
m2
D0
m2
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m2+ = m(K0Sπ+ )2
m2
D0
m2