Introduction to the PWA Isobar TechniqueThe Illinois-Protvino-Munich Program

Quirin Weitzel

Physik-Department E18Technische Universitat Munchen

Physics and Methods in Meson SpectroscopyInt. Joint Workshop CERN-Jefferson Lab-GSI/FAIR

Munich, October 23, 2008

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 1 / 18

1 Motivation and Introduction

2 (Our) PWA Technique

3 Example: Diffractive Meson Production at COMPASS

4 Summary and Outlook

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 2 / 18

Motivation and Introduction

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 3 / 18

Physics Case and Task of the PWA

Hadron spectroscopy: find (all) resonances and determine their properties(like mass M, width Γ, Spin J, Parity P, partial decay widths)

Different Production Mechanisms:

pp annihilation

Diffractive dissociation

Central production

Photo production

...� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �

� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �

π+

Pb

π−− −

π−

π X

Reggeon

)2 System (GeV/c+π-π-πMass of 0 0.5 1 1.5 2 2.5 3

)2E

vent

s / (

5 M

eV/c

0

0.5

1

1.5

2

2.5

3

3.5

310×

(1260)1a

(1320)2a

(1670)2π

COMPASS 2004

Pb+π-π-π →Pb -π2/c 20.1 < t’ < 1.0 GeV

preliminary

Partial Wave Analysis:

Find and disentangle overlappingresonances X

Determine JP from angulardistributions of input events

Take into account exp. acceptance

→ Establish a model to fit the data

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 4 / 18

Physics Case and Task of the PWA

Hadron spectroscopy: find (all) resonances and determine their properties(like mass M, width Γ, Spin J, Parity P, partial decay widths)

Different Production Mechanisms:

pp annihilation

Diffractive dissociation

Central production

Photo production

...� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �

� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �

π+

Pb

π−− −

π−

π X

Reggeon

)2 System (GeV/c+π-π-πMass of 0 0.5 1 1.5 2 2.5 3

)2E

vent

s / (

5 M

eV/c

0

0.5

1

1.5

2

2.5

3

3.5

310×

(1260)1a

(1320)2a

(1670)2π

COMPASS 2004

Pb+π-π-π →Pb -π2/c 20.1 < t’ < 1.0 GeV

preliminary

Partial Wave Analysis:

Find and disentangle overlappingresonances X

Determine JP from angulardistributions of input events

Take into account exp. acceptance

→ Establish a model to fit the data

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 4 / 18

Physics Case and Task of the PWA

Hadron spectroscopy: find (all) resonances and determine their properties(like mass M, width Γ, Spin J, Parity P, partial decay widths)

Different Production Mechanisms:

pp annihilation

Diffractive dissociation

Central production

Photo production

...� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �

� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �

π+

Pb

π−− −

π−

π X

Reggeon

)2 System (GeV/c+π-π-πMass of 0 0.5 1 1.5 2 2.5 3

)2E

vent

s / (

5 M

eV/c

0

0.5

1

1.5

2

2.5

3

3.5

310×

(1260)1a

(1320)2a

(1670)2π

COMPASS 2004

Pb+π-π-π →Pb -π2/c 20.1 < t’ < 1.0 GeV

preliminary

Partial Wave Analysis:

Find and disentangle overlappingresonances X

Determine JP from angulardistributions of input events

Take into account exp. acceptance

→ Establish a model to fit the data

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 4 / 18

The Isobar Model and Decay Amplitudes

S

J PCM ε

target recoil

ε = +: natural

ε =

parity exchange

parity exchange−: unnatural

X L1

2Rππ

π

π

ππ

+

−

−−(beam) (bachelor)

Isobar Model:

How to describe the decay?→ only two-body decaysassumed

L-S coupling

(Known) intermediateresonances: isobars

→ Partial wave: JPCMε[isobar ]L

(π− beam: IG fixed to 1−, C = +1)

Decay Amplitudes ψ(τ,m) τ : phase space, m = mX

Choose a spin formalism: e. g. helicity, Zemach, ... (includes ref. system)

reflectivity basis:

ψεJM(τ,m) = c(M)

[ψJM(τ,m)− εP(−1)J−MψJ(−M)(τ,m)

]Multiply sub-amplitudes for each tree node/isobar→ contain angular distr. (D-funct./sph. harmonics) and isobar description

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 5 / 18

The Isobar Model and Decay Amplitudes

S

J PCM ε

target recoil

ε = +: natural

ε =

parity exchange

parity exchange−: unnatural

X L1

2Rππ

π

π

ππ

+

−

−−(beam) (bachelor)

Isobar Model:

How to describe the decay?→ only two-body decaysassumed

L-S coupling

(Known) intermediateresonances: isobars

→ Partial wave: JPCMε[isobar ]L

(π− beam: IG fixed to 1−, C = +1)

Decay Amplitudes ψ(τ,m) τ : phase space, m = mX

Choose a spin formalism: e. g. helicity, Zemach, ... (includes ref. system)

reflectivity basis:

ψεJM(τ,m) = c(M)

[ψJM(τ,m)− εP(−1)J−MψJ(−M)(τ,m)

]Multiply sub-amplitudes for each tree node/isobar→ contain angular distr. (D-funct./sph. harmonics) and isobar description

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 5 / 18

Spherical Harmonics Y ml (θ, φ)

<(Y 11 )

(rad)θ0 0.5 1 1.5 2 2.5 3

(rad)φ

01

23

45

6

-0.3-0.2-0.1

00.10.20.30.4

<(Y 24 )

(rad)θ0 0.5 1 1.5 2 2.5 3

(rad)φ

01

23

45

6-0.4-0.3-0.2-0.1

00.10.20.30.4

Total intensity of Y 11 and Y 2

4

Real life more complicated!

(rad)θ0 0.5 1 1.5 2 2.5 3

(rad)φ

01

23

45

60

0.020.040.060.08

0.10.120.140.160.18

0.20.220.24

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 6 / 18

(Our) PWA Technique

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 7 / 18

General Ansatz

Model: The total PWA Cross-Section

σ(τ,m) =∑

ε=±1

Nr∑r=1

∣∣∣∣∑i

∑k

C εikr BWk(m,M0 k , Γ0 k) ψ

εi (τ,m)

∣∣∣∣2ε: reflectivity, Nr : rank of spin density matrix

i : different partial waves

k: Breit-Wigner functions (or background): BWk(m,M0 k , Γ0 k)

C : complex production amplitudes

ψ: complex decay amplitudes

Waves with different ε do not interfere (parity conservation)

π− scattering on nucleons: Nr = 2 (spin-flip/non-flip processes)

Practical approach to determine production amplitudes and BW parameters→ task split into two parts: mass-indep. PWA and mass-dep. fit

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 8 / 18

General Ansatz

Model: The total PWA Cross-Section

σ(τ,m) =∑

ε=±1

Nr∑r=1

∣∣∣∣∑i

∑k

C εikr BWk(m,M0 k , Γ0 k) ψ

εi (τ,m)

∣∣∣∣2ε: reflectivity, Nr : rank of spin density matrix

i : different partial waves

k: Breit-Wigner functions (or background): BWk(m,M0 k , Γ0 k)

C : complex production amplitudes

ψ: complex decay amplitudes

Waves with different ε do not interfere (parity conservation)

π− scattering on nucleons: Nr = 2 (spin-flip/non-flip processes)

Practical approach to determine production amplitudes and BW parameters→ task split into two parts: mass-indep. PWA and mass-dep. fit

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 8 / 18

Mass-Independent PWA

Mass-Independent Cross-Section and Spin Density Matrix ρ

σindep(τ,m) =∑

ε=±1

Nr∑r=1

∣∣∣∣∑i

T εirψ

εi (τ,m)

/√∫|ψε

i (τ′,m)|2dτ ′

∣∣∣∣2 , ρεij =

Nr∑r=1

T εirT

ε∗jr

Within (chosen) mass bins the mass-dependence of X is droppedFit parameters are the complex amplitudes T ε

ir → Nr production vectorsFor each mass bin a separate fit is performed

Extended Maximum Likelihood Method

ln L =Nevents∑n=1

lnσindep(τn,mn)−∫ ∫

σindep(τ,m) Acc(τ,m) dτ dm

Nevents: event sample (real data), Acc: Acceptance (from MC)

Fitting Package

“Fumili/Ascoli/Kachaev” fitter: first and second derivatives used

Multiple solutions (∆ ln L ≤ 1) tested → additional error

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 9 / 18

Mass-Independent PWA

Mass-Independent Cross-Section and Spin Density Matrix ρ

σindep(τ,m) =∑

ε=±1

Nr∑r=1

∣∣∣∣∑i

T εirψ

εi (τ,m)

/√∫|ψε

i (τ′,m)|2dτ ′

∣∣∣∣2 , ρεij =

Nr∑r=1

T εirT

ε∗jr

Within (chosen) mass bins the mass-dependence of X is droppedFit parameters are the complex amplitudes T ε

ir → Nr production vectorsFor each mass bin a separate fit is performed

Extended Maximum Likelihood Method

ln L =Nevents∑n=1

lnσindep(τn,mn)−∫ ∫

σindep(τ,m) Acc(τ,m) dτ dm

Nevents: event sample (real data), Acc: Acceptance (from MC)

Fitting Package

“Fumili/Ascoli/Kachaev” fitter: first and second derivatives used

Multiple solutions (∆ ln L ≤ 1) tested → additional error

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 9 / 18

Mass-Independent PWA

Mass-Independent Cross-Section and Spin Density Matrix ρ

σindep(τ,m) =∑

ε=±1

Nr∑r=1

∣∣∣∣∑i

T εirψ

εi (τ,m)

/√∫|ψε

i (τ′,m)|2dτ ′

∣∣∣∣2 , ρεij =

Nr∑r=1

T εirT

ε∗jr

Within (chosen) mass bins the mass-dependence of X is droppedFit parameters are the complex amplitudes T ε

ir → Nr production vectorsFor each mass bin a separate fit is performed

Extended Maximum Likelihood Method

ln L =Nevents∑n=1

lnσindep(τn,mn)−∫ ∫

σindep(τ,m) Acc(τ,m) dτ dm

Nevents: event sample (real data), Acc: Acceptance (from MC)

Fitting Package

“Fumili/Ascoli/Kachaev” fitter: first and second derivatives used

Multiple solutions (∆ ln L ≤ 1) tested → additional error

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 9 / 18

Physics Observables from Mass-Indep. PWA

Output of mass-indep. PWA is spin density matrix in each mass bin

Structure of Spin Density Matrix ρεij =

Nr∑r=1

T εirT

ε∗jr (hermitian)

Block-diagonal in ε

Chung-Trueman parameterization: [S. U. Chung and T. L. Trueman, Phys. Rev. D11, 633]

First r − 1 elements in each T εir put to zero

First non-zero element T εrr purely real

Physics Observables

Intensity of a partial wave: ρεii

Interference between partial waves described by off-diagonal elements ρεij

ρεij = r ε

ij e iφεij and Cohε

ij =rεij√

ρεii ρ

εjj

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 10 / 18

Physics Observables from Mass-Indep. PWA

Output of mass-indep. PWA is spin density matrix in each mass bin

Structure of Spin Density Matrix ρεij =

Nr∑r=1

T εirT

ε∗jr (hermitian)

Block-diagonal in ε

Chung-Trueman parameterization: [S. U. Chung and T. L. Trueman, Phys. Rev. D11, 633]

First r − 1 elements in each T εir put to zero

First non-zero element T εrr purely real

Physics Observables

Intensity of a partial wave: ρεii

Interference between partial waves described by off-diagonal elements ρεij

ρεij = r ε

ij e iφεij and Cohε

ij =rεij√

ρεii ρ

εjj

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 10 / 18

Mass-Dependent Fit

Input: spin density matrix in each mass bin (no events anymore!)

Task: fit (subset of) spin density matrix as function of mass→ find a model to describe PW intensities and their interferences

Decomposition of the mass-dep. Spin Density Matrix

ρεij(m) =

Nr∑r=1

(∑k

C εikrBWk(m)

√∫|ψε

i (τ)|2dτ) (∑

l

C εjlrBWl(m)

√∫|ψε

j (τ)|2dτ)∗

Output: Parameters of BW functions describing the states X

Further Remarks

“BW” functions can also stand for background (like Deck effect)

MINUIT used for fitting: χ2 fit

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 11 / 18

Mass-Dependent Fit

Input: spin density matrix in each mass bin (no events anymore!)

Task: fit (subset of) spin density matrix as function of mass→ find a model to describe PW intensities and their interferences

Decomposition of the mass-dep. Spin Density Matrix

ρεij(m) =

Nr∑r=1

(∑k

C εikrBWk(m)

√∫|ψε

i (τ)|2dτ) (∑

l

C εjlrBWl(m)

√∫|ψε

j (τ)|2dτ)∗

Output: Parameters of BW functions describing the states X

Further Remarks

“BW” functions can also stand for background (like Deck effect)

MINUIT used for fitting: χ2 fit

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 11 / 18

Example: Diffractive MesonProduction at COMPAS

(Selected Results from 2004)

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 12 / 18

Analysis Overview

Event Selection:

190 GeV/c π− beam on Pb

Diffractive trigger

Primary vertex in target with 3outgoing particles (−−+)

Exclusivity: target stays intact

∼ 400 000 events with0.1 < t ′ < 1.0 GeV2/c2

)2 System (GeV/c+π-π-πMass of 0 0.5 1 1.5 2 2.5 3

)2E

vent

s / (

5 M

eV/c

0

0.5

1

1.5

2

2.5

3

3.5

310×

(1260)1a

(1320)2a

(1670)2π

COMPASS 2004

Pb+π-π-π →Pb -π2/c 20.1 < t’ < 1.0 GeV

preliminary

PWA DetailsMass-independent fit:

40MeV/c2 mass bins, 42 waves (lower mass thresholds)Isobars: ρ(770), f2(1270), ρ3(1690), (ππ)s (separated f0(980)), f0(980)Nucleon target ⇒ rank 2, Pomeron exchange ⇒ more ε = +1 wavesLarge t′ range ⇒ t′-dependence of PWs modelled

Mass-dependent fit: 7 waves (all with ε = +1)

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 13 / 18

Analysis Overview

Event Selection:

190 GeV/c π− beam on Pb

Diffractive trigger

Primary vertex in target with 3outgoing particles (−−+)

Exclusivity: target stays intact

∼ 400 000 events with0.1 < t ′ < 1.0 GeV2/c2

)2 System (GeV/c+π-π-πMass of 0 0.5 1 1.5 2 2.5 3

)2E

vent

s / (

5 M

eV/c

0

0.5

1

1.5

2

2.5

3

3.5

310×

(1260)1a

(1320)2a

(1670)2π

COMPASS 2004

Pb+π-π-π →Pb -π2/c 20.1 < t’ < 1.0 GeV

preliminary

PWA DetailsMass-independent fit:

40MeV/c2 mass bins, 42 waves (lower mass thresholds)Isobars: ρ(770), f2(1270), ρ3(1690), (ππ)s (separated f0(980)), f0(980)Nucleon target ⇒ rank 2, Pomeron exchange ⇒ more ε = +1 wavesLarge t′ range ⇒ t′-dependence of PWs modelled

Mass-dependent fit: 7 waves (all with ε = +1)

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 13 / 18

Partial Wave Set for Mass-Indep. Fit (42 Waves)

JPCMε L Isobar π Cut [GeV]

0−+0+ S f0π 1.400−+0+ S (ππ)sπ -0−+0+ P ρπ -1−+1+ P ρπ -1++0+ S ρπ -1++0+ P f2π 1.201++0+ P (ππ)sπ 0.841++0+ D ρπ 1.301++1+ S ρπ -1++1+ P f2π 1.401++1+ P (ππ)sπ 1.401++1+ D ρπ 1.402−+0+ S f2π 1.202−+0+ P ρπ 0.802−+0+ D f2π 1.502−+0+ D (ππ)sπ 0.802−+0+ F ρπ 1.202−+1+ S f2π 1.202−+1+ P ρπ 0.802−+1+ D f2π 1.502−+1+ D (ππ)sπ 1.202−+1+ F ρπ 1.20

JPCMε L Isobar π Cut [GeV]

2++1+ P f2π 1.502++1+ D ρπ -3++0+ S ρ3π 1.503++0+ P f2π 1.203++0+ D ρπ 1.503++1+ S ρ3π 1.503++1+ P f2π 1.203++1+ D ρπ 1.504−+0+ F ρπ 1.204−+1+ F ρπ 1.204++1+ F f2π 1.604++1+ G ρπ 1.64

1−+0− P ρπ -1−+1− P ρπ -1++1− S ρπ -2−+1− S f2π 1.202++0− P f2π 1.302++0− D ρπ -2++1− P f2π 1.30

FLAT

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 14 / 18

Partial Waves used in Mass-Dep. Fit (7 Waves)

JPCMε L Isobar π Cut [GeV]

0−+0+ S f0π 1.400−+0+ S (ππ)sπ -0−+0+ P ρπ -1−+1+ P ρπ -1++0+ S ρπ -1++0+ P f2π 1.201++0+ P (ππ)sπ 0.841++0+ D ρπ 1.301++1+ S ρπ -1++1+ P f2π 1.401++1+ P (ππ)sπ 1.401++1+ D ρπ 1.402−+0+ S f2π 1.202−+0+ P ρπ 0.802−+0+ D f2π 1.502−+0+ D (ππ)sπ 0.802−+0+ F ρπ 1.202−+1+ S f2π 1.202−+1+ P ρπ 0.802−+1+ D f2π 1.502−+1+ D (ππ)sπ 1.202−+1+ F ρπ 1.20

JPCMε L Isobar π Cut [GeV]

2++1+ P f2π 1.502++1+ D ρπ -3++0+ S ρ3π 1.503++0+ P f2π 1.203++0+ D ρπ 1.503++1+ S ρ3π 1.503++1+ P f2π 1.203++1+ D ρπ 1.504−+0+ F ρπ 1.204−+1+ F ρπ 1.204++1+ F f2π 1.604++1+ G ρπ 1.64

1−+0− P ρπ -1−+1− P ρπ -1++1− S ρπ -2−+1− S f2π 1.202++0− P f2π 1.302++0− D ρπ -2++1− P f2π 1.30

FLAT

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 15 / 18

1++0+ρπS and 2−+0+f2πS

)2 System (GeV/c+π-π-πMass of 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2In

tens

ity /

(40

MeV

/c

0

2

4

6

8

10

12

14

16310×

Sπρ+0++1

2/c 20.1 < t’ < 1.0 GeV

COMPASS 2004

Pb+π-π-π →Pb -π

preliminary

)2 System (GeV/c+π-π-πMass of 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2In

tens

ity /

(40

MeV

/c

0

0.5

1

1.5

2

2.5

3

3.5310×

Sπ2f+0-+2

2/c 20.1 < t’ < 1.0 GeV

COMPASS 2004

Pb+π-π-π →Pb -π

preliminary

)2 System (GeV/c+π-π-πMass of 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Phas

e (d

egre

es)

0

50

100

150

200

250

300

350Pb+π-π-π →Pb -π

preliminary

2/c 20.1 < t’ < 1.0 GeV

S)πρ+0++ S - 1π2f+0-+ (2φ∆

COMPASS 2004 BW for a1(1260) + background:

M = (1.256± 0.006 +0.007−0.017) GeV

Γ = (0.366± 0.009 +0.028−0.025) GeV

BW for π2(1670):

M = (1.659± 0.003 +0.024−0.008) GeV

Γ = (0.271± 0.009 +0.022−0.024) GeV

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 16 / 18

2++1+ρπD

)2 System (GeV/c+π-π-πMass of 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2In

tens

ity /

(40

MeV

/c

0

2

4

6

8

10

12

310× Dπρ+1++2

2/c 20.1 < t’ < 1.0 GeV

COMPASS 2004

Pb+π-π-π →Pb -π

preliminary

)2 System (GeV/c+π-π-πMass of 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Phas

e (d

egre

es)

-300

-250

-200

-150

-100

-50

0 Pb+π-π-π →Pb -π

preliminary

2/c 20.1 < t’ < 1.0 GeV

S)πρ+0++ D - 1πρ+1++ (2φ∆

COMPASS 2004

Two Breit-Wigners needed to describe 2++1+ρπD phase motion:

BW1 for a2(1320) + BW2 for a2(1700)

M = (1.321± 0.001 +0.000−0.007) GeV, Γ = (0.110± 0.002 +0.002

−0.015) GeV

a2(1700) parameters fixed to PDG values: M = 1.732 GeV, Γ = 0.194 GeV

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 17 / 18

Summary and Outlook

PWA technique (Illinois-Protvino-Munich Program)

Basic assumption: isobar modelTwo-step procedure: mass-independent PWA and mass-dependent fitPhysics observables derived from spin density matrix

Examples from COMPASS 2004 data

Crosscheck with BNL-E852 Program planned

Different fitting packages could be tried→ A. Caldwell, “Bayesian Fitting - BAT, a new tool”, Friday, 11:20

Include relativistic effects in PWA (decay amplitudes)→ J. Friedrich, “Relativistic Effects in PW Formulation”, Friday, 12:00

THANK YOU!

Q. Weitzel (TUM E18) PWA Isobar Technique Methods in Meson Spectroscopy 18 / 18