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Partial Wave Analysis Lectures at the “School on Hadron Physics”

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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

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)

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
Style 1 Style 2 Style 3 Style 4 Style 5 Style 6 Style 1 Style 2 Style 1 Style 2 Style 3 Style 1 Style 2 Style 3 Style 1 Style 2 Style 3 Intensity I=ΨΨ* Polarangle φ Phase δ Speed dφ/dm Inelasticity η Real Part Imaginary Part Argand Plot

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