The CMU Baryon Amplitude Analysis Program - JLab N* Analysis … · 2006-11-03 · The CMU Baryon Amplitude Analysis Program JLab N∗ Analysis Workshop D. Applegate M. Bellis Z

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


What are we looking for?

Outline





5 Summary



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





1969, Lichtenberg








1969, Lichtenberg






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



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



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





Baryons

Experimentally measure differential cross sectionsPolarization observablesPublish these and let theorists and phenomenologists try to fit to these.Try to describe

πN → πN





BaryonsExperimentally measure differential cross sections

Polarization observablesPublish these and let theorists and phenomenologists try to fit to these.Try to describe

πN → πN





BaryonsExperimentally measure differential cross sectionsPolarization observables

Publish these and let theorists and phenomenologists try to fit to these.Try to describe

πN → πN





BaryonsExperimentally measure differential cross sectionsPolarization observablesPublish these and let theorists and phenomenologists try to fit to these.

Try to describe

πN → πN





BaryonsExperimentally measure differential cross sectionsPolarization observablesPublish these and let theorists and phenomenologists try to fit to these.Try to describe

πN → πN





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.





MesonsAB → XY

X → π′s,K ′s, η′s etc.Dalitz analysisUnbinned likelihood analysis at the amplitude level.






MesonsAB → XYX → π′s,K ′s, η′s etc.

Dalitz analysisUnbinned likelihood analysis at the amplitude level.






MesonsAB → XYX → π′s,K ′s, η′s etc.Dalitz analysis

Unbinned likelihood analysis at the amplitude level.






MesonsAB → XYX → π′s,K ′s, η′s etc.Dalitz analysisUnbinned likelihood analysis at the amplitude level.






MesonsAB → XYX → π′s,K ′s, η′s etc.Dalitz analysisUnbinned likelihood analysis at the amplitude level.




Current analysis model

Experiment

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observables, dσ�

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Theory




Experiment@

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observables, dσ

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Theory




Experiment@

@R

observables, dσ�

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Theory



Proposed analysis model

Experiment

Theory

?

6

Models - 4-vectors




Experiment

Theory

?

6

Models

- 4-vectors




Experiment

Theory

?

6

Models - 4-vectors


Where are we looking?

Outline





5 Summary



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 program


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 program


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 program


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 program


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 program



γp → Σ+K 0

γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000



CMU PWA program



γp → Σ+K 0

γp → (n)π+ > 10,000,000γp → p(π0) > 10,000,000

Currently analyzingUp on deck...



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.




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.




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.




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.




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.




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.





γp → ΛK+

Single decay pathsIsospin filters: access only N∗’s.

Can key on S11 resonances for pη decays.Significant contribution from t−channel







γp → ΛK+

Single decay pathsIsospin filters: access only N∗’s.Can key on S11 resonances for pη decays.

Significant contribution from t−channel







γp → ΛK+

Single decay pathsIsospin filters: access only N∗’s.Can key on S11 resonances for pη decays.Significant contribution from t−channel







γp → ΛK+



Contribution from u−channel processes.

Background under each process.Limited angular information with which to extract resonanceparameters.





γp → ΛK+



Contribution from u−channel processes.Background under each process.

Limited angular information with which to extract resonanceparameters.





γp → ΛK+





How are we looking? Overview example

Outline





5 Summary



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...



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...



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.



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


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.



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


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




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


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


...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




A toy example

)2W (GeV/c1400 1600 1800 2000 2200 2400

b)

µ (σ

0

10

20

30

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60

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100


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.



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


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.



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


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.



A toy example

)2W (GeV/c1400 1600 1800 2000 2200 2400

b)

µ (σ

0

10

20

30

40

50

60

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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.



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


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.



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


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...



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


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 ∆




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


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 ∆




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


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 ∆




A toy example

)2W (GeV/c1400 1600 1800 2000 2200 2400

b)

µ (σ

0

10

20

30

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100


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 ∆




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


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 ∆




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


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 ∆




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


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 ∆




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


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 ∆




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


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 ∆




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


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 ∆




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


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 ∆




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


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.


How are we looking? What physics do we put in?

Outline





5 Summary



Writing our amplitudes

π0

π+

π−

pi

pf

γWhat do we do with this?

Use Rarita-Schwinger formalism.

Calculate ψ as function of...




π0

π+

π−

IJP

pi

pf

γ

E/M L

X

What do we do with this?



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




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?



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



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



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.



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





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





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



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.



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




π0

π+

π−

pi

pf

��

��

��

��

γ

ω

g

γωηg

ppω

η


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.




π0

π+

π−

pi

pf

��

��

��

��

γ

ω

g

γωηg

ppω

η


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.


How are we looking? How do we code up the physics?

Outline





5 Summary



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...



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...



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)




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)




...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));

.

.

.




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)

...






int main()

{

Awesome amp

generation code

return 0;

}


(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)

...






int main()

{

Awesome amp

generation code

return 0;

}


(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)

...




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++


How are we looking? How do we fit the data?

Outline





5 Summary



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.




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




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



Intensity for one wave

Iα = |Ψα(τ)|2

= |g1αg2αψα(τ)|2

=∑mγ

∑mi

∑mf

|g1αg2αψα(τ)|2




Iα = |Ψα(τ)|2


=∑mγ

∑mi

∑mf

|g1αg2αψα(τ)|2




Iα = |Ψα(τ)|2


=∑mγ

∑mi

∑mf

|g1αg2αψα(τ)|2



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+


2,mf =− 1

2(τ)

˛2+

˛g1αg2αψα;mγ=−1,mi =+ 1

2,mf =+ 1

2(τ)

˛2+


2,mf =− 1

2(τ)

˛2+

˛g1αg2αψα;mγ=−1,mi =− 1

2,mf =+ 1

2(τ)

˛2+


2,mf =− 1

2(τ)

˛2+ |non-interfering background term|2





2,mf =+ 1

2(τ)

˛2+

˛g1αg2αψα;mγ=+1,mi =+ 1

2,mf =− 1

2(τ)

˛2+


2,mf =+ 1

2(τ)

˛2+


2,mf =− 1

2(τ)

˛2+


2,mf =+ 1

2(τ)

˛2+


2,mf =− 1

2(τ)

˛2+


2,mf =+ 1

2(τ)

˛2+


2,mf =− 1

2(τ)

˛2

+ |non-interfering background term|2





2,mf =+ 1

2(τ)

˛2+

˛g1αg2αψα;mγ=+1,mi =+ 1

2,mf =− 1

2(τ)

˛2+


2,mf =+ 1

2(τ)

˛2+


2,mf =− 1

2(τ)

˛2+


2,mf =+ 1

2(τ)

˛2+


2,mf =− 1

2(τ)

˛2+


2,mf =+ 1

2(τ)

˛2+


2,mf =− 1

2(τ)

˛2+ |non-interfering background term|2



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.



Writing the code



Constrain couplings across different waves/datasets.

Look at projections of the data after the fit. (dσ)Do systematic checks.




Writing the code



Constrain couplings across different waves/datasets.Look at projections of the data after the fit. (dσ)

Do systematic checks.




Writing the code







Writing the code






How far have we gotten?

Outline





5 Summary



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.



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



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.



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...


Summary

Outline





5 Summary


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


Summary

Summary


We have put togther a complete analysis package which should allow us totackle this comprehensive program.

CMU Machinery

EBAC modelsPPPPPq

CLAS data

��1


CB-ELSA,etc. data

PPPPPi


Summary

Summary


We have both the data and tools to perform a mass-independent, eventbased PWA.

CMU Machinery

EBAC modelsPPPPPq

CLAS data

��1


CB-ELSA,etc. data

PPPPPi


Summary

Summary


Will be able to perform coupled channel analysis.

CMU Machinery

EBAC modelsPPPPPq

CLAS data

��1


CB-ELSA,etc. data

PPPPPi


Summary

Summary


Can naturally apply theory directly to the CLAS data analysis.

CMU Machinery

EBAC modelsPPPPPq

CLAS data

��1


CB-ELSA,etc. data

PPPPPi


Summary

Summary


Would be able to naturally include data from other experiments and mod-els!

CMU Machinery

EBAC modelsPPPPPq

CLAS data

��1


CB-ELSA,etc. data

PPPPPi


Summary

Summary


Generality makes this program extensible to other analysis (GlueX, E852,CLEO Dalitz analysis etc.).

CMU Machinery

EBAC modelsPPPPPq

CLAS data

��1


CB-ELSA,etc. data

PPPPPi


Summary

Summary


Very excited about near-future developments!

CMU Machinery

EBAC modelsPPPPPq

CLAS data

��1


CB-ELSA,etc. data

PPPPPi


Documents

The CMU Baryon Amplitude Analysis Program - JLab N* Analysis … · 2006-11-03 · The CMU Baryon Amplitude Analysis Program JLab N∗ Analysis Workshop D. Applegate M. Bellis Z