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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