1. Partial Wave Analysis Lectures 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
Goals
Wave Approach
Isobar-Model
Level of Detail
Introduction and Concepts
Spin Formalisms
Dynamical Functions
Technical Issues

4. Overview – Spin Formalisms
Zemach Formalism
Canonical Formalism
Helicity Formalism
Moments Analysis
Introduction and Concepts
Spin Formalisms
Dynamical Functions
Technical Issues

5. Overview - Dynamical Functions
Introduction and Concepts
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
Introduction and Concepts
Spin Formalisms
Dynamical Functions
Technical Issues
Coding Amplitudes
Speed is an Issue
Fitting Methods
Caveats
FAQ

7. Header – 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 don‘t 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, n=3
2 Observables
From four vectors 12
Conservation laws -4
Meson masses -3
Free rotation -3
Σ 2
Usual choice
Invariant mass m12
Invariant mass m13
Dalitz plot π1 pp π2 π3

12. J/ψ ® π+π-π0 Angular distributions are easily seen in the Dalitz plot
cosθ -1 +1

13. Phase Space Plot - Dalitz Plot
Q small
Q large
dN ~ (E1dE1) (E2dE2) (E3dE3)/(E1E2E3)
Energy conservation E3 = Etot-E1-E2
Phase space density ρ = dN/dEtot ~ dE1 dE2
Kinetic energies Q=T1+T2+T3
Plot x=(T2-T1)/√3
y=T3-Q/3
Flat, if no dynamics
is involved

14. The first plots  τ/θ-Puzzle
Dalitz applied it first to KL-decays
The former τ/θ 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 pattern due 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. It’s All a Question of Statistics ...
pp ® 3p0 with
100 events
Crystal Barrel 3pi0 Plots

17. It’s All a Question of Statistics ... ...
pp ® 3p0 with
100 events
1000 events

18. It’s All a Question of Statistics ... ... ...
pp ® 3p0 with
100 events
1000 events
10000 events

19. It’s All a Question of Statistics ... ... ... ...
pp ® 3p0 with
100 events
1000 events
10000 events
events

20. Experimental Techniques
Scattering Experiments
πN - N* measurement
πN - meson spectroscopy
E818, AGS, GAMS
pp meson threshold production
Celsius, COSY
pp or πp in the central region
WA76, WA91, WA102
γN – photo production
Cebaf, Mami, Elsa, Graal
“At-rest” Experiments
rest at LEAR
Asterix, Obelix, Crystal Barrel
J/ψ decays
MarkIII,DM2,BES,CLEO-c
ф(1020) decays
Dafne, VEPP
D and Ds decays
FNAL, Babar, Belle

21. Introducing Partial Waves
Schrödinger‘s Equation
Angular Amplitude
Dynamic Amplitude

22. Argand Plot

23. Standard Breit-Wigner
Full circle in the Argand Plot
Phase motion from 0 to π
Argand Plot
Intensity I=ΨΨ*
Phase δ
Speed dφ/dm

24. Breit-Wigner in the Real World
e+e- ® ππ
mππ
ρ-ω

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
Isobar

27. The Full Amplitude For each node an amplitude f(I,I3,s,Ω) 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 ρ0π0 forbidden

29. Header – 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-+)®f2π0
Construct total spin 0 amplitude
Angular
distribution
(Intensity)
A=Af2π x Aππ

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

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
From two-particle state
LS-Coefficients

45. Helicity Decay Amplitudes
From two-particle state
Helicity amplitude

46. Spin Density and Observed Number of Events
To finally calculate the intensity i.e. the number of events observed
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 |j1m1> and |j2m2> to final system <JM|
decay of an initial system |JM> to <j1m1| and <j2m2|
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 J M m1 m2
<j1m1j2m2|JM>

49. Using Clebsch-Gordan Tables, Case 1
1 x 1 2 +2 1 +1
1/2
-1/2 -1
1/6
1/3
2/3
-1/3 -2

50. Using Clebsch-Gordan Tables, Case 2
1 x 1 2 +2 1 +1
1/2
-1/2 -1
1/6
1/3
2/3
-1/3 -2

51. Parity Transformation and Conservation
single particle
two particles
helicity amplitude relations (for P conservation)

52. f2 ® ππ (Ansatz) Initial: f2(1270) IG(JPC) = 0+(2++)
Final: π0 IG(JPC) = 1-(0-+)
Only even angular momenta, since ηf=ηπ2(-1)l
Total spin s=2sπ=0
Ansatz

53. f2 ® ππ (Rates)
Amplitude has to be symmetrized because of the final state particles

54. ω ® π0γ (Ansatz) Initial: ω IG(JPC) = 0-(1--)
Final: π0 IG(JPC) = 1-(0-+)
γ IG(JPC) = 0(1--)
Only odd angular momenta, since ηω=ηπηγ(-1)l
Only photon contributes to total spin s=sπ+sγ
Ansatz

55. ω ® π0γ (Rates)
λγ=±1 do not interfere, λγ=0 does not exist for real photons
Rate depends on density matrix
Choose uniform density matrix as an example

56. f0,2 ® γγ (Ansatz) Initial: f0,2 IG(JPC) = 0+(0,2++)
Final: γ IG(JPC) = 0(1--)
Only even angular momenta, since ηf=ηγ2(-1)l
Total spin s=2sγ=2, l=0,2 (f0), l=0,2,4 (f2)
Ansatz

57. f0,2 ® γγ (cont‘d) Ratio between a00 and a22 is not measurable
Problem even worse for J=2

58. f0,2 ® γγ (cont‘d)
Usual assumption J=λ=2

59. pp (2++) ® ππ Proton antiproton in flight into two pseudo scalars
Initial: pp J,M=0,±1
Final: π IG(JPC) = 1-(0-+)
Ansatz
Problem: d-functions are not orthogonal, if φ is not observed
ambiguities remain in the amplitude – polarization is needed

60. pp ® π0ω Two step process First step pp®π0ω - Second step ω®π0γ
Combine the amplitudes
helicity constant aω,11 factorizes and is unimportant for angular distributions

61. pp (0-+) ® f2π0 Initial: pp IG(JPC) = 1-(0-+)
Final: f2(1270) IG(JPC) = 0+(2++)
π0 IG(JPC) = 1-(0-+)
is only possible from L=2
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 ® 0+0
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 ® J+0

63. Moments Analysis Consider reaction Total differential cross section
expand H
leading to

64. Moments Analysis cont‘d
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KSK+K-

66. Proton-Antiproton Annihilation @ Rest
Atomic initial system
formation at high n, l (n~30)
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 S P D F
n=4
n=3
n=2
n=1
Stark-Effect
S-Wave
P-Wave
(99% of 2P)
Annihilation
ext. Auger-Effect b a L K
(1%)
rad. Transition

67. Initial States @ Rest JPC IG L S 1S0 0-+ pseudo scalar 1-;0+ 3S1 1--
3S1
1--
vector
1+;0- 1
1P1
1+-
axial vector
3P0
0++
scalar
3P1
1++
3P2
2++
tensor
Quantumnumbers
G=(-1)I+L+S
P=(-1)L+1
C=(-1)L+S CP=(-1)2L+S+1
I=0
I=1

68. Proton-Antiproton Annihilation in Flight
scattering process: no 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
H++=0 if L+S-J odd
CP-Invariance
H+-=0 if S=0 and/or J=0
CP transform
CP=(-1)2L+S+1
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 q=0)
odd and even L decouples
4 incoherent sets of coherent amplitudes

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

71. Header – 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 Schrödinger‘s equation
Incoming wave
Solves the equation
solution without interaction
solution with interaction
outgoing
wave
incoming
wave
inelasticity and phase shift

75. T-Matrix from Scattering (cont’d)
wave function
Scattering amplitude
and T-Matrix
Example: ππ-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-<{b}

81. Relativistic Breit-Wigner
Argand Plot
Phase δ
Intensity I=ΨΨ*
By migrating from Schrödinger‘s equation (non-relativistic)
to Klein-Gordon‘s equation (relativistic) the energy term changes
different energy-momentum relation E=p2/m vs. E2=m2c4+p2c2
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 qR=qscale

84. Blatt-Weisskopf Barrier Factors (l=0 to 3)
Usage

85. Resonant daughters Barrier factors 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 <Fl(q)>=∫Fl(q)dBW
Fl(q)~ql
complex even above threshold
meaning of mass and width are mixed up
Im(q)
threshold
Re(q)

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: ππ-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 Strange effects in subdominant channels
Argand Plot
Phase δ
Intensity I=ΨΨ*
Strange effects in subdominant channels
Scalar resonance at 1500 MeV/c2, Γ=100 MeV/c2
All plots show ππ channel
Blue: ππ dominated resonance (Γππ=80 MeV and ΓKK=20 MeV)
Red: KK dominated resonance (ΓKK=80 MeV and Γππ=20 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
2 BW
K-Matrix
FWHM
FWHM

96. Example: 1x2 K-Matrix Nearby Poles
2 BW
K-Matrix
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 Example a0(980) decaying into πη and KK Real Part Argand Plot
BW πη
Flatte πη
Flatte KK
Example
a0(980) decaying into πη and KK
Intensity I=ΨΨ*
Phase δ

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 Γ(m) implies ρ(m)
but below threshold ρ(m) gets complex
because q (breakup-momentum) gets complex,
since m1+m2>m
therefore the real part of the denominator (mass term) changes
and imaginary part (width term) vanishes completely

104. Complex Analysis Revisited (cont’d)
But furthermore for each ρ(m) which is a squareroot, one has two solutions for p>0 or p<0 resp.
But the two values (w=2q/m) have some phase in between and are not identical
So you define a new complex plane for each solution, which are 2n complex planes, called Riemann sheets they 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 ρ(770) in between two thresholds 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
ρ2 smoothly from q2 to –q2

112. Summary Acknowledgements I‘d 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 I’d like to thank R.S. Longacre, from whom I have stolen some paragraphs from his Lecture in Maryland 1991

113. Legend Master Intensity I=ΨΨ* Polarangle φ Phase δ Speed dφ/dm
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Intensity I=ΨΨ*
Polarangle φ
Phase δ
Speed dφ/dm
Inelasticity η
Real Part
Imaginary Part
Argand Plot