Partial Wave Analysis Lectures at the

1
Partial Wave AnalysisLectures at the School on
Hadron Physics

• Klaus Peters
• Ruhr Universität Bochum
• Varenna, June 2004

E. Fermi CLVII Course
2
Overview

• Overview

Introduction and Concepts Spin
Formalisms Dynamical Functions Technical Issues
3
Overview Introduction and Concepts

• Overview

Goals Wave Approach Isobar-Model Level of Detail
Introduction and Concepts Spin
Formalisms Dynamical Functions Technical Issues
4
Overview Spin Formalisms

• Overview

Overview Zemach Formalism Canonical
Formalism Helicity Formalism Moments Analysis
Introduction and Concepts Spin
Formalisms Dynamical Functions Technical Issues
5
Overview - Dynamical Functions

• Overview

Introduction and Concepts Spin
Formalisms Dynamical Functions Technical Issues
Breit-Wigner S-/T-Matrix K-Matrix P/Q-Vector N/D-M
ethod Barrier Factors Interpretation
6
Overview Technical Issues / Fitting

• Overview

Introduction and Concepts Spin
Formalisms Dynamical Functions Technical Issues
Coding Amplitudes Speed is an Issue Fitting
Methods Caveats FAQ
7
Header Introduction and Concepts

• Introduction
• Concepts

Introduction and Concepts Spin
Formalisms Dynamical Functions Technical issues
8
What is the mission ?

• 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

9
Goal

• 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 dont need just a good
  description,
• you need the right one
• Many solutions may look alike but only one is
  right

10
Intermediate State Mixing

• 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)

11
n-Particle Phase space, n3

• 2 Observables
• From four vectors 12
• Conservation laws -4
• Meson masses -3
• Free rotation -3
• S 2
• Usual choice
• Invariant mass m12
• Invariant mass m13

Dalitz plot
p1
pp
p2
p3
12
J/? pp-p0

• Angular distributions are easily seen in the
  Dalitz plot

13
Phase Space Plot - Dalitz Plot
Q small Q large

• dN (E1dE1) (E2dE2) (E3dE3)/(E1E2E3)
• Energy conservation E3 Etot-E1-E2
• Phase space density ? dN/dEtot dE1 dE2
• Kinetic energies QT1T2T3
• Plot x(T2-T1)/v3
• yT3-Q/3

Flat, if no dynamics is involved
14
The first plots ? t/?-Puzzle

• Dalitz applied it first to KL-decays
• The former t/? puzzle with only a few events
• goal was to determine spin and parity
• And he never called them Dalitz plots

15
Interference problem

• PWA
• The phase space diagram in hadron physics shows
  a patterndue to interference and spin effects
• This is the unbiased measurement
• What has to be determined ?
• Analogy Optics ? PWA
• lamps ? level
• slits ? resonances
• positions of slits ? masses
• sizes of slits ? widths
• but only if spins are properly assigned
• bias due to hypothetical spin-parity assumption

• Optics
• Dalitz plot

16
Its All a Question of Statistics ...

• pp 3p0 with
• 100 events

17
Its All a Question of Statistics ... ...

• pp 3p0 with
• 100 events
• 1000 events

18
Its All a Question of Statistics ... ... ...

• pp 3p0 with
• 100 events
• 1000 events
• 10000 events

19
Its All a Question of Statistics ... ... ... ...

• pp 3p0 with
• 100 events
• 1000 events
• 10000 events
• 100000 events

20
Experimental Techniques

• Scattering Experiments
• pN - N measurement
• pN - meson spectroscopy
• E818, E852 _at_ AGS, GAMS
• pp meson threshold production
• Wasa _at_ Celsius, COSY
• pp or pp in the central region
• WA76, WA91, WA102
• ?N photo production
• Cebaf, Mami, Elsa, Graal

• At-rest Experiments
• pN _at_ rest at LEAR
• Asterix, Obelix, Crystal Barrel
• J/? decays
• MarkIII,DM2,BES,CLEO-c
• ?(1020) decays
• Kloe _at_ Dafne, VEPP
• D and Ds decays
• FNAL, Babar, Belle

21
Introducing Partial Waves

• Schrödingers Equation

Angular Amplitude
Dynamic Amplitude
22
Argand Plot
23
Standard Breit-Wigner

• Full circle in the Argand Plot
• Phase motion from 0 to p

Intensity I??
Argand Plot
Phase d
Speed df/dm
24
Breit-Wigner in the Real World

• ee- pp

mpp
?-?
25
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 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 (see last lecture!)

26
Isobar Model

• 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

27
The Full Amplitude

• For each node an amplitude f(I,I3,s,O) is
  obtained.
• The full amplitude is the sum of all nodes.
• Summed over all unobservables

28
Example Isospin Dependence

• pp initial states differ in isospin
• Calculate isospin Clebsch-Gordan
• 1S0 destructive interferences
• 3S1 ?0p0 forbidden

29
Header Spin Formalisms

• Spin
• Formalisms

Introduction and Concepts Spin
Formalisms Dynamical Functions Technical Issues
30
Formalisms on overview

• 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

31
How To Construct a Formalism

• Key steps are
• Definition of single particle states of given
  momentum and spin component (momentum-states),
• Definition of two-particle momentum-states in the
  s-channel center-of-mass system and of amplitudes
  between them,
• Transformation to states and amplitudes of given
  total angular momentum (J-states),
• Symmetry restrictions on the amplitudes,
• Derive Formulae for observable quantities.

32
Zemach Formalism

• For particle with spin S
• traceless tensor of rank S
• with indices

• Similar for orbital angular momentum L

33
Example Zemach pp (0-)f2p0

• Construct total spin 0 amplitude
• Angular
• distribution
• (Intensity)

• AAf2p x App

34
The Original Zemach Paper
35
Spin-Projection Formalisms

• Differ in choice of quantization axis
• Helicity Formalism
• parallel to its own direction of motion
• Transversity Formalism
• the component normal to the scattering plane is
  used
• Canonical (Orbital) Formalism
• the component m in the incident z-direction is
  diagonal

36
Generalized Single Particle State

• In general all single particle states
• are derived from a lorentz transformation
• and the rotation of the basic state
• with the Wigner rotation

37
Properties
Helicity Transversity Canonical
property possibility/simplicity possibility/simplicity possibility/simplicity
partial wave expansion simple complicated complicated
parity conservation no yes yes
crossing relation no good bad
specification of kinematical constraints no yes yes
38
Rotation of States

• Canonical System

• Helicity System

39
Single Particle State

• Canonical
• 1) momentum vector is rotated via z-direction.
  Secondly
• 2) absolute value of the momentum is Lorentz
  boosted along z
• 3) z-axis is rotated to the momentum direction

40
Single Particle State
Helicity 1) z-axis is rotated to the momentum
direction 2) Lorentz Boost Therefore the new
z-axis, z, is parallel to the momentum
41
Two-Particle State

• Canonical
• constructed from two single-particle states
• (back-to-back)

• Couple s and t to S
• Couple L and S to J
• Spherical Harmonics

42
Two-Particle State

• Helicity
• similar procedure
• no recoupling needed
• normalization

43
Completeness and Normalization

• Canonical
• completeness
• normalization

• Helicity
• completeness
• normalization

44
Canonical Decay Amplitudes

• Canonical
• From two-particle state
• LS-Coefficients

45
Helicity Decay Amplitudes

• Helicity
• From two-particle state
• Helicity amplitude

46
Spin Density and Observed Number of Events

• To finally calculate the intensityi.e. the
  number of eventsobserved
• Spin density of the initial state
• Sum over all unobservables
• taking into account

47
Relations Canonical ? Helicity

• Recoupling coefficients
• Start with
• Canonical to Helicity
• Helicity to Canonical

48
Clebsch-Gordan Tables

• Clebsch-Gordan Coefficients are usually tabled in
  a graphical form
• (like in the PDG)
• Two cases
• coupling two initial particles with j1m1gt and
  j2m2gt to final system ltJM
• decay of an initial system JMgt to ltj1m1 and
  ltj2m2
• j1 and j2 do not explicitly appear in the tables
• all values implicitly contain a square root
• Minus signs are meant to be used in front of the
  square root

j1 x j2 j1 x j2 J J
j1 x j2 j1 x j2 M M
m1 m2 ltj1m1j2m2JMgt ltj1m1j2m2JMgt
m1 m2 ltj1m1j2m2JMgt ltj1m1j2m2JMgt
49
Using Clebsch-Gordan Tables, Case 1
1 x 1 1 x 1 2
1 x 1 1 x 1 2 2 1
1 1 1 1 1
1 0 1/2 1/2 2 1 0
0 1 1/2 -1/2 0 0 0
1 -1 1/6 1/2 1/3
0 0 2/3 0 -1/3 2 1
-1 1 1/6 -1/2 1/3 -1 -1
-1 0 1/2 1/2 2
0 -1 1/2 -1/2 -2
-1 -1 1
50
Using Clebsch-Gordan Tables, Case 2
1 x 1 1 x 1 2
1 x 1 1 x 1 2 2 1
1 1 1 1 1
1 0 1/2 1/2 2 1 0
0 1 1/2 -1/2 0 0 0
1 -1 1/6 1/2 1/3
0 0 2/3 0 -1/3 2 1
-1 1 1/6 -1/2 1/3 -1 -1
-1 0 1/2 1/2 2
0 -1 1/2 -1/2 -2
-1 -1 1
51
Parity Transformation and Conservation

• Parity transformation
• single particle
• two particles
• helicity amplitude relations (for P conservation)

52
f2 pp (Ansatz)

• Initial f2(1270) IG(JPC) 0(2)
• Final p0 IG(JPC) 1-(0-)
• Only even angular momenta, since ?f?p2(-1)l
• Total spin s2sp0
• Ansatz

53
f2 pp (Rates)

• Amplitude has to be symmetrized because of the
  final state particles

54
? p0? (Ansatz)

• Initial ? IG(JPC) 0-(1--)
• Final p0 IG(JPC) 1-(0-)
• ? IG(JPC) 0(1--)
• Only odd angular momenta, since ???p??(-1)l
• Only photon contributes to total spin ssps?
• Ansatz

55
? p0? (Rates)

• ??1 do not interfere, ??0 does not exist for
  real photons
• Rate depends on density matrix
• Choose uniform density matrix as an example

56
f0,2 ?? (Ansatz)

• Initial f0,2 IG(JPC) 0(0,2)
• Final ? IG(JPC) 0(1--)
• Only even angular momenta, since ?f??2(-1)l
• Total spin s2s?2, l0,2 (f0), l0,2,4 (f2)
• Ansatz

57
f0,2 ?? (contd)

• Ratio between a00 and a22 is not measurable
• Problem even worse for J2

58
f0,2 ?? (contd)

• Usual assumption J?2

59
pp (2) pp

• Proton antiproton in flight into two pseudo
  scalars
• Initial pp J,M0,1
• Final p IG(JPC) 1-(0-)
• Ansatz
• Problem d-functions are not orthogonal, if f is
  not observed
• ambiguities remain in the amplitude
  polarization is needed

60
pp p0?

• Two step process
• First step ppp0? - Second step ?p0?
• Combine the amplitudes
• helicity constant a?,11 factorizes and is
  unimportant for angular distributions

61
pp (0-) f2p0

• Initial pp IG(JPC) 1-(0-)
• Final f2(1270) IG(JPC) 0(2)
• p0 IG(JPC) 1-(0-)
• is only possible from L2
• Ansatz

62
General Statements

• Flat angular distributions
• General rules for spin 0
• initial state has spin 0
• 0 any
• both final state particles have spin 0
• J 00
• Special rules for isotropic density matrix and
  unobserved azimuth angle
• one final state particle has spin 0 and the
  second carries the same spin as the initial state
• J J0

63
Moments Analysis

• Consider reaction
• Total differential cross section
• expand H
• leading to

64
Moments Analysis contd

• Define now a density tensor
• the d-function products
• can be expanded in spherical harmonics
• and the density matrix gets absorbed in a
  spherical moment

65
Example Where to start in Dalitz plot anlysis

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

D0?KSKK-
66
Proton-Antiproton Annihilation _at_ Rest

• Atomic initial system
• formation at high n, l (n30)
• slow radiative transitions
• de-excitation through collisions (Auger effect)
• Stark mixing of l-levels (Day, Snow, Sucher
  1960)
• Advantages
• JPC varies with target density
• isospin varies with n (d) or p target
• incoherent initial states
• unambiguous PWA possible
• Disadvantages
• phase space very limited
• small kaon yield

67
Initial States _at_ Rest
JPC IG L S
1S0 0- pseudo scalar 1-0 0 0
3S1 1-- vector 10- 0 1
1P1 1- axial vector 10- 1 0
3P0 0 scalar 1-0 1 1
3P1 1 axial vector 1-0 1 1
3P2 2 tensor 1-0 1 1

• Quantumnumbers
• G(-1)ILS
• P(-1)L1
• C(-1)LS CP(-1)2LS1
• I0
• I1

68
Proton-Antiproton Annihilation in Flight

• Annihilation in flight
• scattering processno well defined initial state
• maximum angular momentum rises with energy
• Advantages
• larger phase space
• formation experiments
• Disadvantages
• many waves interfere with each other
• many waves due to large phase space

69
Scattering Amplitudes in Flight (I)

• pp helicity amplitude
• only H and H- exist
• C-Invariance
• H0 if LS-J odd
• CP-Invariance
• H-0 if S0 and/or J0

• CP transform
• CP(-1)2LS1
• S and CP directly correlated
• CP conserved in strong int.
• singlet and triplet decoupled
• C transform
• L and P directly correlated
• C conserved in strong int.(if total charge is
  q0)
• odd and even L decouples
• 4 incoherent sets of coherent amplitudes

70
Scattering Amplitudes in Flight (II)
Singlett odd L JPC L S H H-
1P1 1- 1 0 Yes No
1F3 3- 3 0 Yes No
1G5 5- 5 0 Yes No
Singlett even L JPC L S H H-
1S0 0- 0 0 Yes No
1D2 2- 2 0 Yes No
1G4 4- 4 0 Yes No
Triplett odd L JPC L S H H-
3P0 0 1 1 Yes No
3P1 1 1 1 No Yes
3P2 2 1 1 Yes Yes
3F2 2 3 1 Yes No
3F3 3 3 1 No Yes
3F4 4 3 1 Yes Yes
Triplett even L JPC L S H H-
3S1 1-- 0 1 Yes Yes
3D1 1-- 2 1 Yes Yes
3D2 2-- 2 1 Yes Yes
3D3 3-- 2 1 Yes Yes
71
Header Dynamical Functions

• Dynamical Functions

Introduction and concepts Spin
Formalisms Dynamical Functions Technical issues
72
S-Matrix

• Differential cross section
• Scattering amplitude
• Total scattering cross section

• S-Matrix
• with
• and

73
Harmonic Oscillator

• Free oscillator
• Damped oscillator
• Solution
• External periodic force
• Oscillation strength
• and phase shift
• Lorentz function

74
T-Matrix from Scattering

• Back to Schrödingers equation
• Incoming wave
• Solves the equation
• solution without interaction
• solution with interaction

incoming wave
outgoing wave
inelasticity and phase shift
75
T-Matrix from Scattering (contd)

• Scattering
• wave function
• Scattering amplitude
• and T-Matrix
• Example pp-Scattering
• below 1 GeV/c2

76
(In-)Elastic cross sections and T-Matrix

• Total cross section
• Identify elastic
• and inelastic part

• using the optical theorem

77
Breit-Wigner Function

• Wave function for an unstable particle
• Fourier transformation for E dependence
• Finally our first Breit-Wigner

78
Dressed Resonances T-Matrix Field Theory

• Suppose we have a resonance with mass m0
• We can describe this with a propagator
• But we may have a self-energy term
• leading to

79
T-Matrix Perturbation


...

• We can have an infinite number of loops inside
  our propagator
• every loop involves a coupling b,
• so if b is small, this converges like a geometric
  series

80
T-Matrix Perturbation Retaining Breit-Wigner

• So we get
• and the full amplitude with a dressed
  propagator leads to
• which is again a Breit-Wigner like function, but
  the bare energy E0 has now changed into E0-ltb

81
Relativistic Breit-Wigner
Intensity I??
Phase d
Argand Plot

• By migrating from Schrödingers equation
  (non-relativistic)
• to Klein-Gordons equation (relativistic) the
  energy term changes
• different energy-momentum relation Ep2/m vs.
  E2m2c4p2c2
• The propagators change to sR-s from mR-m

82
Barrier Factors - Introduction

• At low energies, near thresholds
• but is not valid far away from thresholds --
  otherwise the width would explode and the
  integral of the Breit-Wigner diverges
• Need more realistic centrifugal barriers
• known as Blatt-Weisskopf damping factors
• We start with the semi-classical impact parameter
• and use the approximation for the stationary
  solution of the radial differential equation
• with
• we obtain

83
Blatt-Weisskopf Barrier Factors

• The energy dependence is usually parameterized in
  terms of spherical Hankel-Functions
• we define Fl(q) with the
• following features
• Main problem is the choice of the scale parameter
  qRqscale

84
Blatt-Weisskopf Barrier Factors (l0 to 3)

• Usage

85
Barrier factors
Resonant daughters

• Scales and Formulae
• formula was derived from a cylindrical potential
• the scale (197.3 MeV/c) may be different for
  different processes
• valid in the vicinity of the pole
• definition of the breakup-momentum

• Breakup-momentum
• may become complex (sub-threshold)
• set to zero below threshold
• need ltFl(q)gt?Fl(q)dBW
• Fl(q)ql
• complex even above threshold
• meaning of mass and width are mixed up

Im(q)
Re(q)
threshold
86
T-Matrix Unitarity Relations

• Unitarity is a basic feature
• since probability has to be conserved
• T is unitary if S is unitary
• since
• we get in addition

87
T-Matrix Dispersion Relations

• Cauchy Integral on a closed contour
• By choosing proper contours
• and some limits one obtains the dispersion
  relation for Tl(s)
• Satisfying this relation with an arbitrary
• parameterization is extremely difficult
• and is dropped in many approaches

88
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

89
K-Matrix Properties

• T is then easily computed from K
• T and K commute
• K is the Caley transform of S
• Some more properties

90
Example pp-Scattering

• 1 channel

• 2 channel

91
Relativistic Treatment

• So far we did not care about relativistic
  kinematics
• covariant description
• or
• and
• with
• therefore
• and K is changed as well

92
Relativistic Treatment 2 channel

• S-Matrix
• 2 channel T-Matrix
• to be compared with the non-relativistic case

93
K-Matrix Poles

• Now we introduce resonances
• as poles (propagators)
• One may add cij a real polynomial
• of m2 to account for slowly varying background
• (not experimental background!!!)
• Width/Lifetime
• For a single channel and one pole we get

94
Example 1x2 K-Matrix
Intensity I??
Phase d
Argand Plot

• Strange effects in subdominant channels
• Scalar resonance at 1500 MeV/c2, G100 MeV/c2
• All plots show pp channel
• Blue pp dominated resonance (Gpp80 MeV and
  GKK20 MeV)
• Red KK dominated resonance (GKK80 MeV and
  Gpp20 MeV)
• Look at the tiny phase motion in the subdominant
  channel

95
Example 2x1 K-Matrix Overlapping Poles

• two resonances overlapping with different (100/50
  MeV/c2)
• widths are not so dramatic (except the strength)
• The width is basically added

FWHM
FWHM
96
Example 1x2 K-Matrix Nearby Poles

• Two nearby poles (1.27 and 1.5 GeV/c2)
• show nicely the effect of unitarization

97
Example Flatté 1x2 K-Matrix

• 2 channels for a single resonance at the
  threshold of one of the channels
• with
• Leading to the T-Matrix
• and with
• we get

98
Flatté
Real Part
Argand Plot
BW p?
Flatte p?
Flatte KK

• Example
• a0(980) decaying into p? and KK

Intensity I??
Phase d
99
Example K-Matrix Parametrizations

• Au, Morgan and Pennington (1987)
• Amsler et al. (1995)
• Anisovich and Sarantsev (2003)

100
P-Vector Definition

• But in many reactions there is no scattering
  process but a production process, a resonance is
  produced with a certain strength and then decays
• Aitchison (1972)
• with

101
P-Vector Poles

• The resonance poles are constructed as in the
  K-Matrix
• and one may add a polynomial di again
• For a single channel and a single pole
• If the K-Matrix contains fake poles...
• for non s-channel processes modeled in an
  s-channel model
• ...the corresponding poles in P are different

102
Q-Vector

• A different Ansatz with a different picture
  channel n is produced and undergoes final state
  interaction
• For channel 1 in 2 channels

103
Complex Analysis Revisited

• The Breit-Wigner example
• shows, that G(m) implies ?(m)
• but below threshold ?(m) gets complex
• because q (breakup-momentum) gets complex,
• since m1m2gtm
• therefore the real part of the denominator (mass
  term) changes
• and imaginary part (width term) vanishes
  completely

104
Complex Analysis Revisited (contd)

• But furthermore for each ?(m) which is a
  squareroot, one has two solutions for pgt0 or plt0
  resp.
• But the two values (w2q/m) have some phase in
  betweenand are not identical
• So you define a new complex plane for each
  solution,which are 2n complex planes, called
  Riemann sheetsthey are continuously connected.
  The borderlines are called CUTS.

105
Riemann Sheets in a 2 Channel Problem

• Usual definition
• sheet sgn(q1) sgn(q2)
• I
• II -
• III - -
• IV
• This implies for the T-Matrix

• Complex Energy Plane
• Complex Momentum Plane

106
States on Energy Sheets

• Singularities appear naturally where

• Singularities might be
• 1 bound states
• 2 anti-bound
• states
• 3 resonances
• or
• artifacts due to wrong treatment of the model

107
States on Momentum Sheets

• Or in the complex momentum plane
• Singularities might be
• 1 bound states
• 2 anti-bound states
• 3 resonances

108
Left-hand and Right-hand Cuts

• The right hand CUTS (RHC) come from the open
  channels in an n channel problem
• But also exchange processes and other effects
  introduce CUTS on the left-hand side (LHC)

109
N/D Method

• To get the proper behavior for the left-hand cuts
• Use Nl(s) and Dl(s) which are correlated by
  dispersion relations
• An example for this is the work of Bugg and Zhou
  (1993)

110
Nearest Pole Determines Real Axis

• The pole nearest to the real axis
• or more clearly to a point with mass m on the
  real axis
• determines your physics results
• Far away from thresholds this works nicely
• At thresholds, the world is more
• complicated

• While ?(770) in between twothresholds has a
  beautiful shape
• the f0(980) or a0(980) have not

111
Pole and Shadows near Threshold (2 Channel)

• For a real resonance one always obtains poles on
  sheet II and III
• due to symmetries in Tl
• Usually
• To make sure that pole an shadow match and form
  an s-channel resonance, it is mandatory to check
  if the pole on sheets II and III match
• This is done by artificially changing
• ?2 smoothly from q2 to q2

112
Summary
Acknowledgements

• Id like to thank the organizers
• U. Wiedner and T. Bressani
• for their kind invitation to Varenna and for the
  pressure to prepare the lecture and to write it
  down for later use
• I also would like to thank
• S.U. Chung and M.R. Pennington
• for teaching me, what I hopped to have taught
  you!
• and finally Id like to thank R.S. Longacre, from
  whom I have stolen some paragraphs from his
  Lecture in Maryland 1991

113
Legend Master
Intensity I??
Polarangle f
Phase d
Speed df/dm
Inelasticity ?
Real Part
Imaginary Part
Argand Plot