Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
The CMU Baryon Amplitude Analysis ProgramJLab N∗ Analysis Workshop
D. Applegate M. Bellis Z. Krahn C. Meyer M. Williams
Department of PhysicsCarnegie Mellon University
Nov. 4th, 2006
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 1 / 66
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 2 / 66
What are we looking for?
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 3 / 66
What are we looking for?
Attacking the missing baryons problem
Big motivation is the missing baryons issue.Two competing explanations:
1969, Lichtenberg
Di-quark coupling amongst the 3 constituent quarks.Removes a degree of freedom when calculating masses.Very good agreement with observed resonances.
1980, Koniuk and Isgur
1980, Harmonic oscillator model with QCD-inspired correctionsCalculate decay couplingsFound that many missing states couple relatively weakly to Nπ!But much of experimental data is Nπ → Nπ scattering
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 4 / 66
What are we looking for?
Attacking the missing baryons problem
Big motivation is the missing baryons issue.Two competing explanations:
1969, Lichtenberg
Di-quark coupling amongst the 3 constituent quarks.Removes a degree of freedom when calculating masses.Very good agreement with observed resonances.
1980, Koniuk and Isgur
1980, Harmonic oscillator model with QCD-inspired correctionsCalculate decay couplingsFound that many missing states couple relatively weakly to Nπ!But much of experimental data is Nπ → Nπ scattering
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 4 / 66
What are we looking for?
Attacking the missing baryons problem
Big motivation is the missing baryons issue.Two competing explanations:
1969, Lichtenberg
Di-quark coupling amongst the 3 constituent quarks.Removes a degree of freedom when calculating masses.Very good agreement with observed resonances.
1980, Koniuk and Isgur
1980, Harmonic oscillator model with QCD-inspired correctionsCalculate decay couplingsFound that many missing states couple relatively weakly to Nπ!But much of experimental data is Nπ → Nπ scattering
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 4 / 66
What are we looking for?
Predictions - Capstick and Roberts (1993)
Width Branching fraction (%)State (MeV/c2) Nπ ∆π Nρ Nη Nω ΛK ΣK
N 12
+(1880) 150 5 49 3 18 14 0 9
N 12
+(1975) 50 8 47 14 0 22 3 1
N 32
+(1870) 190 20 12 2 26 11 0 26
N 32
+(1910) 390 0 75 3 0 17 0 2
N 32
+(1950) 140 12 43 11 0 28 3 1
N 32
+(2030) 90 4 57 15 0 16 1 0
N 52
+(1995) 270 1 89 2 0 3 0 0
N 52
+(2000) 190 0 51 33 4 8 0 0
N 72
+(1835) 50 13 53 3 21 6 0 2
∆ 12
+(1985) 310 5 63 20 0 0 0 3
∆ 32
+(1880) 220 5 44 25 0 0 0 5
∆ 52
+(1990) 350 0 56 10 0 0 0 0
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 5 / 66
What are we looking for?
Predictions - Capstick and Roberts (1993)
Width Branching fraction (%)State (MeV/c2) Nπ ∆π Nρ Nη Nω ΛK ΣK
N 12
+(1880) 150 5 49 3 18 14 0 9
N 12
+(1975) 50 8 47 14 0 22 3 1
N 32
+(1870) 190 20 12 2 26 11 0 26
N 32
+(1910) 390 0 75 3 0 17 0 2
N 32
+(1950) 140 12 43 11 0 28 3 1
N 32
+(2030) 90 4 57 15 0 16 1 0
N 52
+(1995) 270 1 89 2 0 3 0 0
N 52
+(2000) 190 0 51 33 4 8 0 0
N 72
+(1835) 50 13 53 3 21 6 0 2
∆ 12
+(1985) 310 5 63 20 0 0 0 3
∆ 32
+(1880) 220 5 44 25 0 0 0 5
∆ 52
+(1990) 350 0 56 10 0 0 0 0
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 6 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
BaryonsExperimentally measure differential cross sectionsPolarization observablesPublish these and let theorists and phenomenologists try to fit to these.Try to describe
πN → πN
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 7 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
Baryons
Experimentally measure differential cross sectionsPolarization observablesPublish these and let theorists and phenomenologists try to fit to these.Try to describe
πN → πN
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 7 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
BaryonsExperimentally measure differential cross sections
Polarization observablesPublish these and let theorists and phenomenologists try to fit to these.Try to describe
πN → πN
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 7 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
BaryonsExperimentally measure differential cross sectionsPolarization observables
Publish these and let theorists and phenomenologists try to fit to these.Try to describe
πN → πN
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 7 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
BaryonsExperimentally measure differential cross sectionsPolarization observablesPublish these and let theorists and phenomenologists try to fit to these.
Try to describe
πN → πN
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 7 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
BaryonsExperimentally measure differential cross sectionsPolarization observablesPublish these and let theorists and phenomenologists try to fit to these.Try to describe
πN → πN
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 7 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
Mesons
AB → XYX → π′s,K ′s, η′s etc.Dalitz analysisUnbinned likelihood analysis at the amplitude level.
We are applying the meson approach to our baryon analysis.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 8 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
MesonsAB → XY
X → π′s,K ′s, η′s etc.Dalitz analysisUnbinned likelihood analysis at the amplitude level.
We are applying the meson approach to our baryon analysis.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 8 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
MesonsAB → XYX → π′s,K ′s, η′s etc.
Dalitz analysisUnbinned likelihood analysis at the amplitude level.
We are applying the meson approach to our baryon analysis.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 8 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
MesonsAB → XYX → π′s,K ′s, η′s etc.Dalitz analysis
Unbinned likelihood analysis at the amplitude level.
We are applying the meson approach to our baryon analysis.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 8 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
MesonsAB → XYX → π′s,K ′s, η′s etc.Dalitz analysisUnbinned likelihood analysis at the amplitude level.
We are applying the meson approach to our baryon analysis.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 8 / 66
What are we looking for?
Purpose of CMU program
Historically, meson and and baryon analysis have involved differinganalysis procedures and different applications of PWA techniques.
MesonsAB → XYX → π′s,K ′s, η′s etc.Dalitz analysisUnbinned likelihood analysis at the amplitude level.
We are applying the meson approach to our baryon analysis.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 8 / 66
What are we looking for?
Current analysis model
Experiment
@@R
observables, dσ�
��
Theory
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 9 / 66
What are we looking for?
Current analysis model
Experiment@
@R
observables, dσ
���
Theory
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 9 / 66
What are we looking for?
Current analysis model
Experiment@
@R
observables, dσ�
��
Theory
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 9 / 66
What are we looking for?
Proposed analysis model
Experiment
Theory
?
6
Models - 4-vectors
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 10 / 66
What are we looking for?
Proposed analysis model
Experiment
Theory
?
6
Models
- 4-vectors
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 10 / 66
What are we looking for?
Proposed analysis model
Experiment
Theory
?
6
Models - 4-vectors
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 10 / 66
Where are we looking?
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 11 / 66
Where are we looking?
CMU PWA program
Analyze reactions with no Nπ (e.g. photoproduction)
Available channels from CLAS data.Channel Secondary decays/isobars Events (before fiducial cuts)
γp → pπ+π− ∆++π−,∆0π+, pρ0 >10,000,000γp → pω ω → π+, π−, (π0) >10,000,000γp → pη η → π+, π−, (π0) 1,500,000γp → pη′ η′ → π+, π−, (η) 500,000γp → ΛK+ Λ → pπ− 1,700,000γp → Σ0K+
γp → Σ+K 0
γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 12 / 66
Where are we looking?
CMU PWA program
Analyze reactions with no Nπ (e.g. photoproduction)
Available channels from CLAS data.Channel Secondary decays/isobars Events (before fiducial cuts)γp → pπ+π− ∆++π−,∆0π+, pρ0 >10,000,000
γp → pω ω → π+, π−, (π0) >10,000,000γp → pη η → π+, π−, (π0) 1,500,000γp → pη′ η′ → π+, π−, (η) 500,000γp → ΛK+ Λ → pπ− 1,700,000γp → Σ0K+
γp → Σ+K 0
γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 12 / 66
Where are we looking?
CMU PWA program
Analyze reactions with no Nπ (e.g. photoproduction)
Available channels from CLAS data.Channel Secondary decays/isobars Events (before fiducial cuts)γp → pπ+π− ∆++π−,∆0π+, pρ0 >10,000,000γp → pω ω → π+, π−, (π0) >10,000,000
γp → pη η → π+, π−, (π0) 1,500,000γp → pη′ η′ → π+, π−, (η) 500,000γp → ΛK+ Λ → pπ− 1,700,000γp → Σ0K+
γp → Σ+K 0
γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 12 / 66
Where are we looking?
CMU PWA program
Analyze reactions with no Nπ (e.g. photoproduction)
Available channels from CLAS data.Channel Secondary decays/isobars Events (before fiducial cuts)γp → pπ+π− ∆++π−,∆0π+, pρ0 >10,000,000γp → pω ω → π+, π−, (π0) >10,000,000γp → pη η → π+, π−, (π0) 1,500,000γp → pη′ η′ → π+, π−, (η) 500,000
γp → ΛK+ Λ → pπ− 1,700,000γp → Σ0K+
γp → Σ+K 0
γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 12 / 66
Where are we looking?
CMU PWA program
Analyze reactions with no Nπ (e.g. photoproduction)
Available channels from CLAS data.Channel Secondary decays/isobars Events (before fiducial cuts)γp → pπ+π− ∆++π−,∆0π+, pρ0 >10,000,000γp → pω ω → π+, π−, (π0) >10,000,000γp → pη η → π+, π−, (π0) 1,500,000γp → pη′ η′ → π+, π−, (η) 500,000γp → ΛK+ Λ → pπ− 1,700,000γp → Σ0K+
γp → Σ+K 0
γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 12 / 66
Where are we looking?
CMU PWA program
Analyze reactions with no Nπ (e.g. photoproduction)
Available channels from CLAS data.Channel Secondary decays/isobars Events (before fiducial cuts)γp → pπ+π− ∆++π−,∆0π+, pρ0 >10,000,000γp → pω ω → π+, π−, (π0) >10,000,000γp → pη η → π+, π−, (π0) 1,500,000γp → pη′ η′ → π+, π−, (η) 500,000γp → ΛK+ Λ → pπ− 1,700,000γp → Σ0K+
γp → Σ+K 0
γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 12 / 66
Where are we looking?
CMU PWA program
Analyze reactions with no Nπ (e.g. photoproduction)
Available channels from CLAS data.Channel Secondary decays/isobars Events (before fiducial cuts)γp → pπ+π− ∆++π−,∆0π+, pρ0 >10,000,000γp → pω ω → π+, π−, (π0) >10,000,000γp → pη η → π+, π−, (π0) 1,500,000γp → pη′ η′ → π+, π−, (η) 500,000γp → ΛK+ Λ → pπ− 1,700,000γp → Σ0K+
γp → Σ+K 0
γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000
Currently analyzingUp on deck...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 13 / 66
Where are we looking?
Pros/Cons of each channel
γp → pπ+π−
Multiple isobars: ∆++π−,∆0π+, pρ
Accesses both N∗’s and ∆’s.Many known s−channel resonances:D13(1520),D13(1700),D15(1675),F15(1680), etc.Contribution from t−channel ρ production (Pomeron exchange.)Contribution from t−channel π production (π exchange).Contribution from contact (Kroll-Ruderman) term.However, ability to measure final state in different ways (reconstructingany missing track) give you a systematic check on the acceptance ofthe detector.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 14 / 66
Where are we looking?
Pros/Cons of each channel
γp → pπ+π−
Multiple isobars: ∆++π−,∆0π+, pρAccesses both N∗’s and ∆’s.
Many known s−channel resonances:D13(1520),D13(1700),D15(1675),F15(1680), etc.Contribution from t−channel ρ production (Pomeron exchange.)Contribution from t−channel π production (π exchange).Contribution from contact (Kroll-Ruderman) term.However, ability to measure final state in different ways (reconstructingany missing track) give you a systematic check on the acceptance ofthe detector.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 14 / 66
Where are we looking?
Pros/Cons of each channel
γp → pπ+π−
Multiple isobars: ∆++π−,∆0π+, pρAccesses both N∗’s and ∆’s.Many known s−channel resonances:D13(1520),D13(1700),D15(1675),F15(1680), etc.
Contribution from t−channel ρ production (Pomeron exchange.)Contribution from t−channel π production (π exchange).Contribution from contact (Kroll-Ruderman) term.However, ability to measure final state in different ways (reconstructingany missing track) give you a systematic check on the acceptance ofthe detector.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 14 / 66
Where are we looking?
Pros/Cons of each channel
γp → pπ+π−
Multiple isobars: ∆++π−,∆0π+, pρAccesses both N∗’s and ∆’s.Many known s−channel resonances:D13(1520),D13(1700),D15(1675),F15(1680), etc.Contribution from t−channel ρ production (Pomeron exchange.)Contribution from t−channel π production (π exchange).Contribution from contact (Kroll-Ruderman) term.
However, ability to measure final state in different ways (reconstructingany missing track) give you a systematic check on the acceptance ofthe detector.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 14 / 66
Where are we looking?
Pros/Cons of each channel
γp → pπ+π−
Multiple isobars: ∆++π−,∆0π+, pρAccesses both N∗’s and ∆’s.Many known s−channel resonances:D13(1520),D13(1700),D15(1675),F15(1680), etc.Contribution from t−channel ρ production (Pomeron exchange.)Contribution from t−channel π production (π exchange).Contribution from contact (Kroll-Ruderman) term.However, ability to measure final state in different ways (reconstructingany missing track) give you a systematic check on the acceptance ofthe detector.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 14 / 66
Where are we looking?
Pros/Cons of each channel
γp → pωγp → pη/η′
γp → ΛK+
Single decay paths
Isospin filters: access only N∗’s.Can key on S11 resonances for pη decays.Significant contribution from t−channel
ω production (π exchange.)η/η′ production (ρ/ω exchange.)
Contribution from u−channel processes.Background under each process.Limited angular information with which to extract resonanceparameters.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 15 / 66
Where are we looking?
Pros/Cons of each channel
γp → pωγp → pη/η′
γp → ΛK+
Single decay pathsIsospin filters: access only N∗’s.
Can key on S11 resonances for pη decays.Significant contribution from t−channel
ω production (π exchange.)η/η′ production (ρ/ω exchange.)
Contribution from u−channel processes.Background under each process.Limited angular information with which to extract resonanceparameters.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 15 / 66
Where are we looking?
Pros/Cons of each channel
γp → pωγp → pη/η′
γp → ΛK+
Single decay pathsIsospin filters: access only N∗’s.Can key on S11 resonances for pη decays.
Significant contribution from t−channel
ω production (π exchange.)η/η′ production (ρ/ω exchange.)
Contribution from u−channel processes.Background under each process.Limited angular information with which to extract resonanceparameters.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 15 / 66
Where are we looking?
Pros/Cons of each channel
γp → pωγp → pη/η′
γp → ΛK+
Single decay pathsIsospin filters: access only N∗’s.Can key on S11 resonances for pη decays.Significant contribution from t−channel
ω production (π exchange.)η/η′ production (ρ/ω exchange.)
Contribution from u−channel processes.Background under each process.Limited angular information with which to extract resonanceparameters.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 15 / 66
Where are we looking?
Pros/Cons of each channel
γp → pωγp → pη/η′
γp → ΛK+
Single decay pathsIsospin filters: access only N∗’s.Can key on S11 resonances for pη decays.Significant contribution from t−channel
ω production (π exchange.)η/η′ production (ρ/ω exchange.)
Contribution from u−channel processes.
Background under each process.Limited angular information with which to extract resonanceparameters.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 15 / 66
Where are we looking?
Pros/Cons of each channel
γp → pωγp → pη/η′
γp → ΛK+
Single decay pathsIsospin filters: access only N∗’s.Can key on S11 resonances for pη decays.Significant contribution from t−channel
ω production (π exchange.)η/η′ production (ρ/ω exchange.)
Contribution from u−channel processes.Background under each process.
Limited angular information with which to extract resonanceparameters.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 15 / 66
Where are we looking?
Pros/Cons of each channel
γp → pωγp → pη/η′
γp → ΛK+
Single decay pathsIsospin filters: access only N∗’s.Can key on S11 resonances for pη decays.Significant contribution from t−channel
ω production (π exchange.)η/η′ production (ρ/ω exchange.)
Contribution from u−channel processes.Background under each process.Limited angular information with which to extract resonanceparameters.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 15 / 66
How are we looking? Overview example
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 16 / 66
How are we looking? Overview example
What are we going to?
Goal is to perform an event-based, energy-independent amplitudeanalysis.
A lot of stuff goes into this.
In order to orient ourselves, we take a look a at a cartoon example ofan ideal procedure...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 17 / 66
How are we looking? Overview example
What are we going to?
Goal is to perform an event-based, energy-independent amplitudeanalysis.
A lot of stuff goes into this.
In order to orient ourselves, we take a look a at a cartoon example ofan ideal procedure...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 17 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
)+
21 -
-
23 (φ ∆
Total cross section for γp → pπ+π− asmeasured in previous experiments.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 18 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
)+
21 -
-
23 (φ ∆
First step: describe the physics in a singleW-bin.
Let the fit find optimal mix of physicsprocesses which describes the data in thissmall energy range.
We do not put in shapes(e.g.Breit-Wigner) for the resonances forwhich we are looking.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 19 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
)+
21 -
-
23 (φ ∆
Dalitz plot data shows structure...
0
50
100
150
200
250
4/c2 GeV2)+πM(p
1.2 1.4 1.6 1.8 2 2.2
4/c2
GeV
2 )- πM
(p
1.2
1.4
1.6
1.8
2
2.2
4/c2 GeV2)+π vs M(p 4/c2 GeV2)-πM(p 4/c2 GeV2)+π vs M(p 4/c2 GeV2)-πM(p
...which is not seen in the acceptedMonte Carlo.
0
20
40
60
80
100
120
140
4/c2 GeV2)+πM(p
1.2 1.4 1.6 1.8 2 2.2
4/c2
GeV
2 )- πM
(p
1.2
1.4
1.6
1.8
2
2.2
4/c2 GeV2)+π vs M(p 4/c2 GeV2)-πM(p 4/c2 GeV2)+π vs M(p 4/c2 GeV2)-πM(p
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 20 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
)+
21 -
-
23 (φ ∆
Dalitz plot data shows structure...
0
50
100
150
200
250
4/c2 GeV2)+πM(p
1.2 1.4 1.6 1.8 2 2.2
4/c2
GeV
2 )- πM
(p
1.2
1.4
1.6
1.8
2
2.2
4/c2 GeV2)+π vs M(p 4/c2 GeV2)-πM(p 4/c2 GeV2)+π vs M(p 4/c2 GeV2)-πM(p
...but which we do get out of the fitresults.
0
50
100
150
200
250
4/c2 GeV2)+πM(p
1.2 1.4 1.6 1.8 2 2.2
4/c2
GeV
2 )- πM
(p
1.2
1.4
1.6
1.8
2
2.2
4/c2 GeV2)+π vs M(p 4/c2 GeV2)-πM(p 4/c2 GeV2)+π vs M(p 4/c2 GeV2)-πM(p
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 21 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
)+
21 -
-
23 (φ ∆
Fit results allow calculation of differentialand total cross sections.
) GeV^2-π+π - γ-t (0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
diffe
rent
ial y
ield
(ar
bitr
ary
units
)
0
50
100
150
200
250
Saphir W=1416-1480
CMU W=1420-1430
CMU W=1430-1440
CMU W=1440-1450
CMU W=1450-1460
CMU W=1460-1470
CMU W=1470-1480
Prelim
inary
) GeV^2-π+π - γ-t (0 0.2 0.4 0.6 0.8 1
diffe
rent
ial y
ield
(ar
bitr
ary
units
)
0
50
100
150
200
250
Saphir W=1542-1603
CMU W=1550-1560
CMU W=1560-1570
CMU W=1570-1580
CMU W=1580-1590
CMU W=1590-1600
Prelim
inary
The differential cross sections are not theend results of this analysis.
In a sense though, we get them for free.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 22 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
)+
21 -
-
23 (φ ∆
Fit results allow calculation of differentialand total cross sections.
) GeV^2-π+π - γ-t (0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
diffe
rent
ial y
ield
(ar
bitr
ary
units
)
0
50
100
150
200
250
Saphir W=1416-1480
CMU W=1420-1430
CMU W=1430-1440
CMU W=1440-1450
CMU W=1450-1460
CMU W=1460-1470
CMU W=1470-1480
Prelim
inary
) GeV^2-π+π - γ-t (0 0.2 0.4 0.6 0.8 1
diffe
rent
ial y
ield
(ar
bitr
ary
units
)
0
50
100
150
200
250
Saphir W=1542-1603
CMU W=1550-1560
CMU W=1560-1570
CMU W=1570-1580
CMU W=1580-1590
CMU W=1590-1600
Prelim
inary
The differential cross sections are not theend results of this analysis.
In a sense though, we get them for free.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 22 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Really want to look at contributions ofthe various physics processes in our fit
In this toy example, we show
contributions from
Non-resonant termγp → 1
2
− → ∆π
γp → 32
+ → ∆πCan also look at the phase difference of12
−and 3
2
+terms.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 23 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
We go to some other bin and perform anindependent fit with the same.
While we may constrain parameters in ournon-resonant terms across W-bins, keepin mind we do not constrain this for thewaves which could describe resonantprocesses.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 24 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
We go to some other bin and perform anindependent fit with the same.
While we may constrain parameters in ournon-resonant terms across W-bins, keepin mind we do not constrain this for thewaves which could describe resonantprocesses.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 24 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 25 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 26 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 27 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 28 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 29 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 30 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 31 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 32 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 33 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 34 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 35 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
Repeat this across the full energy range...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 36 / 66
How are we looking? Overview example
A toy example
)2W (GeV/c1400 1600 1800 2000 2200 2400
b)
µ (σ
0
10
20
30
40
50
60
70
80
90
100
Previous experiments
Non-resonant term
term+
21
term-
23
1400 1600 1800 2000 2200 2400-1.5
-1
-0.5
0
0.5
1
)+
21 -
-
23 (φ ∆
If there are waves which describe resonantprocesses, this should show up in intensityand phase differences even though thisinformation was not put into the fit apriori.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 37 / 66
How are we looking? What physics do we put in?
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 38 / 66
How are we looking? What physics do we put in?
Writing our amplitudes
π0
π+
π−
pi
pf
γWhat do we do with this?
Use Rarita-Schwinger formalism.
Calculate ψ as function of...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 39 / 66
How are we looking? What physics do we put in?
Writing our amplitudes
π0
π+
π−
IJP
pi
pf
γ
E/M L
X
What do we do with this?
Use Rarita-Schwinger formalism.
Calculate ψ as function of...
X: isobar (e.g. ω, η, η′)Intermediate state:I: isospinJ: spinP: parityE/M: production multipoleL: final state orbital angularmomentum
Also a function of initial/finalspin-projections
mγ ,mi ,mf
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 40 / 66
How are we looking? What physics do we put in?
Writing our amplitudes
π0
π+
π−
IJP
pi
pf
γ
E/M L
X
We do not put in Breit-Wignershapes for the IJP intermediate state
What do we do with this?
Use Rarita-Schwinger formalism.
Calculate ψ as function of...
X: isobar (e.g. ω, η, η′)Intermediate state:I: isospinJ: spinP: parityE/M: production multipoleL: final state orbital angularmomentum
Also a function of initial/finalspin-projections
mγ ,mi ,mf
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 41 / 66
How are we looking? What physics do we put in?
Fit parameters
π0
π+
π−
IJP
pi
pf
������������������
������������������
������������������
������������������
γ
E/M L
X
α β
We do not put in Breit-Wignershapes for the IJP intermediate state
Fit α
Ratio of E/M multipole.Production phase - in futuremass dependant analysis, thismaps onto a Breit-Wignerphase.Indexed by I, J, P
Fit β
Decay strength and overallscaleIndexed by I, J, P, isobar, L
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 42 / 66
How are we looking? What physics do we put in?
t− and u−channel diagrams
π0
π+
π−
pi
pf
g1
������������������
������������������
ω
2
f(s,t)
γ
g
We have our pick of propagator(Regge, Feynman, etc.)
Can write out our u-channelterms in the same way.
Fit the t−channel coupling forthe higher photon energies, andthen fix the value everywhere.
Can put in couplings directlyfrom previous measurements.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 43 / 66
How are we looking? What physics do we put in?
For example: γp → pω
Can have the same production couplings, but different decay couplings fordifferent L’s in final state.
π0
π+
π−
p
γ
p
ω =
π0
π+
π−
pi
pf
2_1
2_1 −
������������������
������������������
������������������
������������������
γ
L=0
ω
E1α β
0
+
π0
π+
π−
pi
pf
2_1
2_1 −
������������������
������������������
������������������
������������������
γ
L=2
ω
E1α
2β
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 44 / 66
How are we looking? What physics do we put in?
For example: γp → pω
Can have the same production couplings, but different decay couplings fordifferent L’s in final state.
π0
π+
π−
p
γ
p
ω = π0
π+
π−
pi
pf
2_1
2_1 −
������������������
������������������
������������������
������������������
γ
L=0
ω
E1α β
0
+
π0
π+
π−
pi
pf
2_1
2_1 −
������������������
������������������
������������������
������������������
γ
L=2
ω
E1α
2β
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 44 / 66
How are we looking? What physics do we put in?
For example: γp → pω
Can have the same production couplings, but different decay couplings fordifferent L’s in final state.
π0
π+
π−
p
γ
p
ω = π0
π+
π−
pi
pf
2_1
2_1 −
������������������
������������������
������������������
������������������
γ
L=0
ω
E1α β
0
+
π0
π+
π−
pi
pf
2_1
2_1 −
������������������
������������������
������������������
������������������
γ
L=2
ω
E1α
2β
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 44 / 66
How are we looking? What physics do we put in?
Coupled channel
Most immediate application of this is to apply to coupled channelanalysis.
Production is same coupling...only decay coupling varies
2π production (∆++π−,∆0π+, pρ0)
Currently in the process of analyzing η/η′ in this way.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 45 / 66
How are we looking? What physics do we put in?
t−channel diagrams across channels
π0
π+
π−
pi
pf
������������������
������������������
������������������
������������������
γ
η
ω
g
γωηg
ppη
Access to multiple channels lets us usedifferent analysis in conjunction with oneanother.
For example, given ω production with η
exchange, there are two couplings that
are fit in the t-channel process.
gγωη
gppη
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 46 / 66
How are we looking? What physics do we put in?
t−channel diagrams across channels
π0
π+
π−
pi
pf
������������������
������������������
������������������
������������������
γ
ω
g
γωηg
ppω
η
Access to multiple channels lets us usedifferent analysis in conjunction with oneanother.
When we analyze η production with ω
exchange, there is a common coupling
with the ω production.
gγωη
gppω
Right now, we try to measure these in onechannel, and put in by hand to the other.
Next step is to fit the datasetssimultaneously.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 47 / 66
How are we looking? What physics do we put in?
t−channel diagrams across channels
π0
π+
π−
pi
pf
������������������
������������������
������������������
������������������
γ
ω
g
γωηg
ppω
η
Access to multiple channels lets us usedifferent analysis in conjunction with oneanother.
When we analyze η production with ω
exchange, there is a common coupling
with the ω production.
gγωη
gppω
Right now, we try to measure these in onechannel, and put in by hand to the other.
Next step is to fit the datasetssimultaneously.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 47 / 66
How are we looking? How do we code up the physics?
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 48 / 66
How are we looking? How do we code up the physics?
Amplitude generation
Requires tensor algebra
CPU intensiveL`
µ1...µ`(p1, p2) is an `th-rank tensor
Need general, fast code.
C++ classes
Matrix, TensorDiracGamma γµ : The Dirac Gamma matriciesDiracSpinor uµ1µ2...µS−1/2
(p,m): Spin-S spinors, half-integer spin
PolVector εµ1µ2...µS : Spin-S polarization vectors, integer spinLeviCivitaTensor εµνρσ : Totally anti-symmetric Levi-Civita tensor
OrbitalTensor L(`)µ1µ2...µ`
: Orbital angular momentum tensors...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 49 / 66
How are we looking? How do we code up the physics?
Amplitude generation
Requires tensor algebra
CPU intensiveL`
µ1...µ`(p1, p2) is an `th-rank tensor
Need general, fast code.
C++ classes
Matrix, TensorDiracGamma γµ : The Dirac Gamma matriciesDiracSpinor uµ1µ2...µS−1/2
(p,m): Spin-S spinors, half-integer spin
PolVector εµ1µ2...µS : Spin-S polarization vectors, integer spinLeviCivitaTensor εµνρσ : Totally anti-symmetric Levi-Civita tensor
OrbitalTensor L(`)µ1µ2...µ`
: Orbital angular momentum tensors...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 49 / 66
How are we looking? How do we code up the physics?
Coding the processes
Start with some process...
ηγ
p p
JP
...write it out using R-S convariant formalism...
Aγp→ 1
2
−→pη∼ umpf
(ppf)P
12 (P)γµγ
5umpi(ppi )ε
µmγ
(pγ)Θ(P)
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 50 / 66
How are we looking? How do we code up the physics?
Coding the processes
Start with some process...
ηγ
p p
JP
...write it out using R-S convariant formalism...
Aγp→ 1
2
−→pη∼ umpf
(ppf)P
12 (P)γµγ
5umpi(ppi )ε
µmγ
(pγ)Θ(P)
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 50 / 66
How are we looking? How do we code up the physics?
Coding the processes
...we can turn this directly into usable C++ code.
Aγp→ 1
2−→pη
∼ umpf(ppf )P
12 (P)γµγ
5umpi(ppi )ε
µmγ
(pγ)Θ(P)
DiracSpinor _u_f; ///< final nucleon spinor
DiracSpinor _u_i; ///< initial nucleon spinor
PolVector _eps_g; ///< photon polarization vector
LeviCivitaTensor _lct; ///< Levi-Civita tensor
DiracGamma _gamma; ///< Dirac gamma matricies
vector<int> _Ls; ///< list of ALL L’s that will be used
vector<OrbitalTensor> _Lf; ///< Lf[L] will be rank-L orbital tensor
.
.
.
Li[l].SetP4(p4_phot,p4_targ);
_m_jp_cm[mg_ind] = psi.Projector() | (Li[l]*_gamma)*(_gamma*_eps_g(m_g));
.
.
.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 51 / 66
How are we looking? How do we code up the physics?
Coding the processes
Take 4-vectors of tracksmeasured in CLAS...
...pass them into our code...
int main()
{
Awesome amp
generation code
return 0;
}
...and get out the amplitudefor these events for a givenprocess.
(0.298813,0.694998)(-0.840490,-0.701032)(-0.941817,0.453170)(1.102081,-0.109626)(-1.055170,-0.492774)(0.647787,0.015414)(0.570040,0.570075)
(-0.075149,-0.643757)(-0.012921,-0.835250)(-0.286855,0.344333)(-0.002072,0.812462)(-0.638055,-0.125474)
...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 52 / 66
How are we looking? How do we code up the physics?
Coding the processes
Take 4-vectors of tracksmeasured in CLAS...
...pass them into our code...
int main()
{
Awesome amp
generation code
return 0;
}
...and get out the amplitudefor these events for a givenprocess.
(0.298813,0.694998)(-0.840490,-0.701032)(-0.941817,0.453170)(1.102081,-0.109626)(-1.055170,-0.492774)(0.647787,0.015414)(0.570040,0.570075)
(-0.075149,-0.643757)(-0.012921,-0.835250)(-0.286855,0.344333)(-0.002072,0.812462)(-0.638055,-0.125474)
...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 52 / 66
How are we looking? How do we code up the physics?
Coding the processes
Take 4-vectors of tracksmeasured in CLAS...
...pass them into our code...
int main()
{
Awesome amp
generation code
return 0;
}
...and get out the amplitudefor these events for a givenprocess.
(0.298813,0.694998)(-0.840490,-0.701032)(-0.941817,0.453170)(1.102081,-0.109626)(-1.055170,-0.492774)(0.647787,0.015414)(0.570040,0.570075)
(-0.075149,-0.643757)(-0.012921,-0.835250)(-0.286855,0.344333)(-0.002072,0.812462)(-0.638055,-0.125474)
...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 52 / 66
How are we looking? How do we code up the physics?
Coding the processes
This code makes things much easier on the whole process.
Calculation no longer relies on simplifying the original amplitude orboosting and working out dependance on particular differential crosssections.
We can take almost any processes from any model and apply directlyto the data.
http://www-meg.phys.cmu.edu/pwa/wiki-qft++
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 53 / 66
How are we looking? How do we fit the data?
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 54 / 66
How are we looking? How do we fit the data?
Maximum likelihood method
Currently our fits primarily use an event-based, Maximum Likelihoodprocedure.
Essential for pπ+π− channel where you have multiple isobars.
Perhaps not quite as necessary for ω or η, η′ channels where there islimited physics information (e.g. cos(θ)CM).
We have recently done some preliminary fits, fitting to differential crosssections for these channels.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 55 / 66
How are we looking? How do we fit the data?
Maximum likelihood method
Construct likelihood function from the intensity
L ∝[nn
n!e−n
] n∏i
I (τi )∫I (τ)η(τ)dτ
...
lnL =n∑i
ln
[∑αα′
g1αg∗1α′g2αg∗2α′ψα(τ)ψ∗α′(τ)
]
− n∑αα′
g1αg∗1α′g2αg∗2α′
1
NAMC
∑iAMC
ψα(τ)ψ∗α′(τ)
α - index representing the waves (IJP ,E/M, L...)
τ - the kinematics of the event
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 56 / 66
How are we looking? How do we fit the data?
Maximum likelihood method
Construct likelihood function from the intensity
L ∝[nn
n!e−n
] n∏i
I (τi )∫I (τ)η(τ)dτ
...
lnL =n∑i
ln
[∑αα′
g1αg∗1α′g2αg∗2α′ψα(τ)ψ∗α′(τ)
]
− n∑αα′
g1αg∗1α′g2αg∗2α′
1
NAMC
∑iAMC
ψα(τ)ψ∗α′(τ)
α - index representing the waves (IJP ,E/M, L...)
τ - the kinematics of the event
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 56 / 66
How are we looking? How do we fit the data?
Intensity for one wave
Iα = |Ψα(τ)|2
= |g1αg2αψα(τ)|2
=∑mγ
∑mi
∑mf
|g1αg2αψα(τ)|2
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 57 / 66
How are we looking? How do we fit the data?
Intensity for one wave
Iα = |Ψα(τ)|2
= |g1αg2αψα(τ)|2
=∑mγ
∑mi
∑mf
|g1αg2αψα(τ)|2
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 57 / 66
How are we looking? How do we fit the data?
Intensity for one wave
Iα = |Ψα(τ)|2
= |g1αg2αψα(τ)|2
=∑mγ
∑mi
∑mf
|g1αg2αψα(τ)|2
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 57 / 66
How are we looking? How do we fit the data?
Intensity for one wave - spins
Iα =˛g1αg2αψα;mγ=+1,mi =+ 1
2,mf =+ 1
2(τ)
˛2
+˛g1αg2αψα;mγ=+1,mi =+ 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=+1,mi =− 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=+1,mi =− 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =+ 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =+ 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =− 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =− 1
2,mf =− 1
2(τ)
˛2+ |non-interfering background term|2
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 58 / 66
How are we looking? How do we fit the data?
Intensity for one wave - spins
Iα =˛g1αg2αψα;mγ=+1,mi =+ 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=+1,mi =+ 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=+1,mi =− 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=+1,mi =− 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =+ 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =+ 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =− 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =− 1
2,mf =− 1
2(τ)
˛2
+ |non-interfering background term|2
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 58 / 66
How are we looking? How do we fit the data?
Intensity for one wave - spins
Iα =˛g1αg2αψα;mγ=+1,mi =+ 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=+1,mi =+ 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=+1,mi =− 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=+1,mi =− 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =+ 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =+ 1
2,mf =− 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =− 1
2,mf =+ 1
2(τ)
˛2+
˛g1αg2αψα;mγ=−1,mi =− 1
2,mf =− 1
2(τ)
˛2+ |non-interfering background term|2
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 58 / 66
How are we looking? How do we fit the data?
Writing the code
What is needed out of the fitting code?Easily construct our likelihood function.
Define who interferes with whom.Add and subtract waves.Fix couplings to known values, or zero out entirely.
Constrain couplings across different waves/datasets.Look at projections of the data after the fit. (dσ)Do systematic checks.
We have a suite of packages utilizing C++, Ruby and XML thathandles all of the above.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 59 / 66
How are we looking? How do we fit the data?
Writing the code
What is needed out of the fitting code?Easily construct our likelihood function.
Define who interferes with whom.Add and subtract waves.Fix couplings to known values, or zero out entirely.
Constrain couplings across different waves/datasets.
Look at projections of the data after the fit. (dσ)Do systematic checks.
We have a suite of packages utilizing C++, Ruby and XML thathandles all of the above.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 59 / 66
How are we looking? How do we fit the data?
Writing the code
What is needed out of the fitting code?Easily construct our likelihood function.
Define who interferes with whom.Add and subtract waves.Fix couplings to known values, or zero out entirely.
Constrain couplings across different waves/datasets.Look at projections of the data after the fit. (dσ)
Do systematic checks.
We have a suite of packages utilizing C++, Ruby and XML thathandles all of the above.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 59 / 66
How are we looking? How do we fit the data?
Writing the code
What is needed out of the fitting code?Easily construct our likelihood function.
Define who interferes with whom.Add and subtract waves.Fix couplings to known values, or zero out entirely.
Constrain couplings across different waves/datasets.Look at projections of the data after the fit. (dσ)Do systematic checks.
We have a suite of packages utilizing C++, Ruby and XML thathandles all of the above.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 59 / 66
How are we looking? How do we fit the data?
Writing the code
What is needed out of the fitting code?Easily construct our likelihood function.
Define who interferes with whom.Add and subtract waves.Fix couplings to known values, or zero out entirely.
Constrain couplings across different waves/datasets.Look at projections of the data after the fit. (dσ)Do systematic checks.
We have a suite of packages utilizing C++, Ruby and XML thathandles all of the above.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 59 / 66
How far have we gotten?
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 60 / 66
How far have we gotten?
Where are we? γp → pπ+π−
Can compare measured differential yields. PRELIMINARY!Black is SAPHIR and blues are CLAS.W=1416-1480 Mev/c2 W=1542-1603 Mev/c2
yielddt t(γ − (π+π−))
) GeV^2-π+π - γ-t (0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
diffe
rent
ial y
ield
(ar
bitr
ary
units
)
0
50
100
150
200
250
Saphir W=1416-1480
CMU W=1420-1430
CMU W=1430-1440
CMU W=1440-1450
CMU W=1450-1460
CMU W=1460-1470
CMU W=1470-1480
Prelim
inary
) GeV^2-π+π - γ-t (0 0.2 0.4 0.6 0.8 1
diffe
rent
ial y
ield
(ar
bitr
ary
units
)
0
50
100
150
200
250
Saphir W=1542-1603
CMU W=1550-1560
CMU W=1560-1570
CMU W=1570-1580
CMU W=1580-1590
CMU W=1590-1600
Prelim
inary
Our binning is finer and we get pretty good agreement.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 61 / 66
How far have we gotten?
γp → pω: Differential Cross Sections
We can compareour results toprevious CLASresults:Battaglieri, et al
We have 21 binsoverlaping their 4W bins(2.62<W<2.87)
Very goodagreement
Notice that thecross sectionappears to bedominated bynon-resonant(t−,u−channel)processes
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 62 / 66
How far have we gotten?
η/η′ 52
−PWA prod. strength and phase results
Production strength of the 52
−partial
wave for the eta and etaprime
eta phase diff.
etaprime phase diff.
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 63 / 66
How far have we gotten?
η/η′ 52
−PWA prod. strength and phase results
For η and η′ we have started extracting coupling constants!
Will soon be coupling this analysis with the ω where they share manycouplings.
Stay tuned...
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 64 / 66
Summary
Outline
1 What are we looking for?
2 Where are we looking?
3 How are we looking?Overview exampleWhat physics do we put in?How do we code up the physics?How do we fit the data?
4 How far have we gotten?
5 Summary
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 65 / 66
Summary
Summary
Non-strange baryon spectroscopy very promising...but not easy.
CMU Machinery
EBAC modelsPPPPPq
CLAS data
�����1
FSU,etc. models�����)
CB-ELSA,etc. data
PPPPPi
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 66 / 66
Summary
Summary
Non-strange baryon spectroscopy very promising...but not easy.
We have put togther a complete analysis package which should allow us totackle this comprehensive program.
CMU Machinery
EBAC modelsPPPPPq
CLAS data
�����1
FSU,etc. models�����)
CB-ELSA,etc. data
PPPPPi
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 66 / 66
Summary
Summary
Non-strange baryon spectroscopy very promising...but not easy.
We have both the data and tools to perform a mass-independent, eventbased PWA.
CMU Machinery
EBAC modelsPPPPPq
CLAS data
�����1
FSU,etc. models�����)
CB-ELSA,etc. data
PPPPPi
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 66 / 66
Summary
Summary
Non-strange baryon spectroscopy very promising...but not easy.
Will be able to perform coupled channel analysis.
CMU Machinery
EBAC modelsPPPPPq
CLAS data
�����1
FSU,etc. models�����)
CB-ELSA,etc. data
PPPPPi
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 66 / 66
Summary
Summary
Non-strange baryon spectroscopy very promising...but not easy.
Can naturally apply theory directly to the CLAS data analysis.
CMU Machinery
EBAC modelsPPPPPq
CLAS data
�����1
FSU,etc. models�����)
CB-ELSA,etc. data
PPPPPi
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 66 / 66
Summary
Summary
Non-strange baryon spectroscopy very promising...but not easy.
Would be able to naturally include data from other experiments and mod-els!
CMU Machinery
EBAC modelsPPPPPq
CLAS data
�����1
FSU,etc. models�����)
CB-ELSA,etc. data
PPPPPi
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 66 / 66
Summary
Summary
Non-strange baryon spectroscopy very promising...but not easy.
Generality makes this program extensible to other analysis (GlueX, E852,CLEO Dalitz analysis etc.).
CMU Machinery
EBAC modelsPPPPPq
CLAS data
�����1
FSU,etc. models�����)
CB-ELSA,etc. data
PPPPPi
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 66 / 66
Summary
Summary
Non-strange baryon spectroscopy very promising...but not easy.
Very excited about near-future developments!
CMU Machinery
EBAC modelsPPPPPq
CLAS data
�����1
FSU,etc. models�����)
CB-ELSA,etc. data
PPPPPi
CMU PWA Group (CMU) CMU Baryon Amplitude Analysis Nov. ’06 66 / 66