The complete experiment problem of pseudoscalarmeson photoproduction in a truncated partial wave
analysis
- Numeric investigations -
Yannick Wunderlich
HISKP, University of Bonn
01.07.2014
Definition of the TPWA problem
For low-energy processes: Truncate the partial wave expansion of the fullspin amplitudes at some finite `max, e.g.
F1(W , θ) =`max∑
=0
[`M`+ + E`+]P
′`+1 (cos θ) + [(`+ 1)M`− + E`−]P
′`−1 (cos θ)
,
and insert this truncated expansion into the polarization observablesΩα (W , θ) , α = 1, . . . , 16
of pseudoscalar meson photoproduction.
Y. Wunderlich Complete experiment in a TPWA
Definition of the TPWA problem
For low-energy processes: Truncate the partial wave expansion of the fullspin amplitudes at some finite `max, e.g.
F1(W , θ) =`max∑
=0
[`M`+ + E`+]P
′`+1 (cos θ) + [(`+ 1)M`− + E`−]P
′`−1 (cos θ)
,
and insert this truncated expansion into the polarization observablesΩα (W , θ) , α = 1, . . . , 16
of pseudoscalar meson photoproduction.
Truncated Partial Wave Analysis
Ωα (W , θ) = sinβα θ[aα0 (W ) + aα1 (W ) cos θ + aα2 (W ) cos2 θ + . . .
]= sinβα θ
2`max+γα∑k=0
aαk (W ) cosk θ,
aαk (W ) = 〈M(W )|Cαk |M(W )〉 , |M (W )〉 = (E`± (W ) ,M`± (W ))T .
→ How many and which observables have to be measured in order touniquely solve for the multipoles E`± (W ) ,M`± (W )?
Y. Wunderlich Complete experiment in a TPWA
Complete sets of observables
Study of the theoretical ambiguities of the group S observables(dσdΩ
)0,Σ,P,T
according to [A. S. Omelaenko (1981)] (see also
[Wunderlich/Beck/Tiator (2014)])
200 300 400 500 600
-4
-2
0
2
4
EΓ @MeVD
H±Φ1±
Φ2L@r
adD
MAID2007
Example: `max = 1
γp → π0pI.
I.
II.
II.
Results of Ambiguity diagrams:
I. the double ambiguity can bepredicted for all orders in `max
and for all energies Eγ
II. accidential ambiguities canoccur randomly in each energybin, but cannot be predicted
Y. Wunderlich Complete experiment in a TPWA
Complete sets of observables
Study of the theoretical ambiguities of the group S observables(dσdΩ
)0,Σ,P,T
according to [A. S. Omelaenko (1981)] (see also
[Wunderlich/Beck/Tiator (2014)])
200 300 400 500 600
-4
-2
0
2
4
EΓ @MeVD
H±Φ1±
Φ2L@r
adD
MAID2007
Example: `max = 1
γp → π0pI.
I.
II.
II.
Results of Ambiguity diagrams:
I. the double ambiguity can bepredicted for all orders in `max
and for all energies Eγ
II. accidential ambiguities canoccur randomly in each energybin, but cannot be predicted
→ Double polarization observables capable of resolving the ambiguities:
BT : F ,G, BR: Ox ′ ,Oz ′ ,Cx ′ ,Cz ′, T R: Tx ′ ,Tz ′ , Lx ′ , Lz ′→ Examples of complete sets: σ0,Σ,T ,P,F & σ0,Σ,T ,P,G→ Can these predictions be verified using numerical TPWA fits?
Y. Wunderlich Complete experiment in a TPWA
Details on the multipole Fit procedure I
Two step method:
1. Fit the angular distributions of observables, parametrized by
Ωα (W , θ) =2`max+βα+γα∑
k=βα
(aL)αk (W )Pβα
k (cos θ)
⇒ Angular fit parameters(aFitL
)αk
- Absorb sinβα θ factors into the fitting functions Pβα
k (cos θ)
- Pβα
k (cos θ) have the advantage of being orthogonal for cos θ ∈ [−1, 1]
Y. Wunderlich Complete experiment in a TPWA
Details on the multipole Fit procedure I
Two step method:
1. Fit the angular distributions of observables, parametrized by
Ωα (W , θ) =2`max+βα+γα∑
k=βα
(aL)αk (W )Pβα
k (cos θ)
⇒ Angular fit parameters(aFitL
)αk
- Absorb sinβα θ factors into the fitting functions Pβα
k (cos θ)
- Pβα
k (cos θ) have the advantage of being orthogonal for cos θ ∈ [−1, 1]
2. Minimize the chisquare functional:
χ2 =∑α,k
((aFitL
)αk− (aL)αk (M`)
)2=∑α,k
((aFitL
)αk− 〈M`| (CL)αk |M`〉
)2
using the MATHEMATICA method
FindMinimum[χ2 (M`) , Re [E0+] , (x1)0 , . . . , Im [M`max−] , (yn)0
]and varying the real and imaginary parts of the (possibly phaseconstrained) multipoles in the fit.
Y. Wunderlich Complete experiment in a TPWA
Details on the multipole Fit procedure I
Two step method:
1. Fit the angular distributions of observables, parametrized by
Ωα (W , θ) =2`max+βα+γα∑
k=βα
(aL)αk (W )Pβα
k (cos θ)
⇒ Angular fit parameters(aFitL
)αk
- Absorb sinβα θ factors into the fitting functions Pβα
k (cos θ)
- Pβα
k (cos θ) have the advantage of being orthogonal for cos θ ∈ [−1, 1]
2. Minimize the chisquare functional:
χ2 =∑α,k
((aFitL
)αk− (aL)αk (M`)
)2=∑α,k
((aFitL
)αk− 〈M`| (CL)αk |M`〉
)2
using the MATHEMATICA method
FindMinimum[χ2 (M`) , Re [E0+] , (x1)0 , . . . , Im [M`max−] , (yn)0
]and varying the real and imaginary parts of the (possibly phaseconstrained) multipoles in the fit.
Problem: How to constrain the start values of the Fit parameters?Y. Wunderlich Complete experiment in a TPWA
Details on the multipole fit procedure II
Remark: The approach presented in the following is presumably containedimplicitly in works such as [Sandorfi, Hoblit, Kamano and Lee (2011)]. However,until now I have not found it described explicitly in the literature.
Y. Wunderlich Complete experiment in a TPWA
Details on the multipole fit procedure II
Remark: The approach presented in the following is presumably containedimplicitly in works such as [Sandorfi, Hoblit, Kamano and Lee (2011)]. However,until now I have not found it described explicitly in the literature.
Ansatz: Use the total cross section σ (W ). Example: ` ≤ `max = 1,
phase constraint Im[E0+
]= 0 & Re
[E0+
]> 0:
σ(W ) ≈ 4π qk
(Re[E0+
]2
+ 6Re[E1+
]2
+ 6 Im[E1+
]2
+ 2Re[M1+
]2
+2 Im[M1+
]2
+ Re[M1−
]2
+ Im[M1−
]2 )
Y. Wunderlich Complete experiment in a TPWA
Details on the multipole fit procedure II
Remark: The approach presented in the following is presumably containedimplicitly in works such as [Sandorfi, Hoblit, Kamano and Lee (2011)]. However,until now I have not found it described explicitly in the literature.
Ansatz: Use the total cross section σ (W ). Example: ` ≤ `max = 1,
phase constraint Im[E0+
]= 0 & Re
[E0+
]> 0:
σ(W ) ≈ 4π qk
(Re[E0+
]2
+ 6Re[E1+
]2
+ 6 Im[E1+
]2
+ 2Re[M1+
]2
+2 Im[M1+
]2
+ Re[M1−
]2
+ Im[M1−
]2 )σ (W ) constrains the intervals of the multipoles:
Re[E0+
]∈[
0,√
kqσ(W )
4π
], . . ., Im
[M1−
]∈[−√
kqσ(W )
4π ,√
kqσ(W )
4π
]The total cross section, being quadratic form in the multipoles, alsodefines an ellipsoid in the multipole space.
Y. Wunderlich Complete experiment in a TPWA
Generation of start values for FindMinimum
1. The total cross section σ(W )constrains the (8`max− 1)-dimensionalmultipole space M`.
Re[E
0+
]
M` \ Re[E0+]
Y. Wunderlich Complete experiment in a TPWA
Generation of start values for FindMinimum
1. The total cross section σ(W )constrains the (8`max− 1)-dimensionalmultipole space M`.
2. σ(W ) defines an (8`max − 2)-dimensional ellipsoid in M`.
Re[E
0+
]
M` \ Re[E0+]
Y. Wunderlich Complete experiment in a TPWA
Generation of start values for FindMinimum
1. The total cross section σ(W )constrains the (8`max− 1)-dimensionalmultipole space M`.
2. σ(W ) defines an (8`max − 2)-dimensional ellipsoid in M`.
3. Solutions to the TPWA problem lie onthe ellipsoid defined by σ(W ).
Re[E
0+
]
M` \ Re[E0+]
Y. Wunderlich Complete experiment in a TPWA
Generation of start values for FindMinimum
1. The total cross section σ(W )constrains the (8`max− 1)-dimensionalmultipole space M`.
2. σ(W ) defines an (8`max − 2)-dimensional ellipsoid in M`.
3. Solutions to the TPWA problem lie onthe ellipsoid defined by σ(W ).
4. The start values for theFindMinimum-Fit are chosenrandomly on the σ(W )-ellipsoid.
⇒ Monte Carlo sampling of themultipole space.
Re[E
0+
]
M` \ Re[E0+]
Y. Wunderlich Complete experiment in a TPWA
Generation of start values for FindMinimum
5. An attempt was made to use theχ2 (M`) for the generation of thestart values.⇒ Clustering of start configurationsnear the minima (cf. [Sandorfi, Hoblit,
Kamano and Lee (2011)]).
However, this approach has not yetbeen usable due to interminablecalculation times (usingMATHEMATICA).
Re[E
0+
]
M` \ Re[E0+]
Y. Wunderlich Complete experiment in a TPWA
Generation of start values for FindMinimum
6. A FindMinimum-minimization isperformed for each of the randomlygenerated start configurations.
⇒ NMC = # of M.C. startconfigurations
= # of (possibly redundant)solutions R
e[E
0+
]
M` \ Re[E0+]
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data I
Fit of group S observables σ0,Σ,T ,P generated using MAID2007multipoles (γp → π0p) up to `max = 1 (Fit `max = 1, NMC = 1000):
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data I
Fit of group S observables σ0,Σ,T ,P generated using MAID2007multipoles (γp → π0p) up to `max = 1 (Fit `max = 1, NMC = 1000):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data I
Fit of group S observables σ0,Σ,T ,P generated using MAID2007multipoles (γp → π0p) up to `max = 1 (Fit `max = 1, NMC = 1000):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2DAmb/χ
2Best ≈ 1
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data I
Fit of group S observables σ0,Σ,T ,P generated using MAID2007multipoles (γp → π0p) up to `max = 1 (Fit `max = 1, NMC = 1000):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2DAmb/χ
2Best ≈ 1
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data I
Fit of group S observables σ0,Σ,T ,P generated using MAID2007multipoles (γp → π0p) up to `max = 1 (Fit `max = 1, NMC = 1000):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Best χ2 solutionOriginal MAID/Double
Ambiguity
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data II
Fit of observables σ0,Σ,T ,P,G generated using MAID2007 multipolesup to `max = 1 (Fit `max = 1, NMC = 1000):
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data II
Fit of observables σ0,Σ,T ,P,G generated using MAID2007 multipolesup to `max = 1 (Fit `max = 1, NMC = 1000):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data II
Fit of observables σ0,Σ,T ,P,G generated using MAID2007 multipolesup to `max = 1 (Fit `max = 1, NMC = 1000):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data II
Fit of observables σ0,Σ,T ,P,G generated using MAID2007 multipolesup to `max = 1 (Fit `max = 1, NMC = 1000):
200 300 400 500 600 700 80001234567
Re[ E
0+
] [10−
3mπ
+]
200 300 400 500 600 700 800
0
5
10
15
20
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
200 300 400 500 600 700 800
-1.0
-0.5
0.0
Re[ E
1+
]
200 300 400 500 600 700 800-20
-15
-10
-5
0
5
Im[ M
1+
]
200 300 400 500 600 700 800
-0.50.00.51.01.52.02.5
Im[ M
1−]
200 300 400 500 600 700 800-1.0
-0.5
0.0
0.5
Im[ E
1+
]
200 300 400 500 600 700 8000.00.51.01.52.02.53.0
Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Fits to truncated MAID theory data II
Fit of observables σ0,Σ,T ,P,G generated using MAID2007 multipolesup to `max = 1 (Fit `max = 1, NMC = 1000):
200 300 400 500 600 700 80001234567
Re[ E
0+
] [10−
3mπ
+]
200 300 400 500 600 700 800
0
5
10
15
20
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
200300400500600700800
-1.0
-0.5
0.0
Re[ E
1+
]
200 300 400 500 600 700 800-20-15-10
-505
Im[ M
1+
]
200300400500600700800
-0.50.00.51.01.52.02.5
Im[ M
1−]
200300400500600700800-1.0
-0.5
0.0
0.5
Im[ E
1+
]
200 300 400 500 600 700 8000.00.51.01.52.02.53.0
Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Original MAID solution
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,G generated using full MAID 2007 (Fit`max = 1, NMC = 1000):
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,G generated using full MAID 2007 (Fit`max = 1, NMC = 1000):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,G generated using full MAID 2007 (Fit`max = 1, NMC = 1000):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,G generated using full MAID 2007 (Fit`max = 1, NMC = 1000):
200 300 400 500 600 700 8000
5
10
15
20
Re[ E
0+
] [10−
3mπ
+]
200 300 400 500 600 700 800
-15-10
-505
1015
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
200 300 400 500 600 700 800-2
-1
0
1
2
3
Re[ E
1+
]
200 300 400 500 600 700 800-4
-2
0
2
4
6
Im[ M
1+
]
200 300 400 500 600 700 800
-4-2
0246
Im[ M
1−]
200 300 400 500 600 700 800-1.5-1.0-0.5
0.00.51.01.5
Im[ E
1+
]
200 300 400 500 600 700 800-10
-5
0
5
Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,G generated using full MAID 2007 (Fit`max = 1, NMC = 1000):
200 300 400 500 600 700 8000
5
10
15
20
Re[ E
0+
] [10−
3mπ
+]
200 300 400 500 600 700 800
-10
0
10
20
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
200 300 400 500 600 700 800-2
-1
0
1
2
3
Re[ E
1+
]
200 300 400 500 600 700 800-20-15-10
-505
Im[ M
1+
]
200 300 400 500 600 700 800
-4-2
0246
Im[ M
1−]
200300400500600700800-1.5-1.0-0.5
0.00.51.01.5
Im[ E
1+
]
200 300 400 500 600 700 800-10
-5
0
5
Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Original MAID solution
Y. Wunderlich Complete experiment in a TPWA
Effect of higher multipoles in lower (aL)αk
∆ resonance: ELABγ ' 340MeV → |M`>1|∣∣∣M∆
`≤1
∣∣∣ = |M`>1||M1+| < |E2−|
|M1+| ' 0.04
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
6
T[ µb sr
]
cos θ-1.0 -0.5 0.0 0.5 1.0
-1.0-0.5
0.00.51.01.52.0
G[ µb sr
]cos θ
Y. Wunderlich Complete experiment in a TPWA
Effect of higher multipoles in lower (aL)αk
∆ resonance: ELABγ ' 340MeV → |M`>1|∣∣∣M∆
`≤1
∣∣∣ = |M`>1||M1+| < |E2−|
|M1+| ' 0.04
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
6
T[ µb sr
]
cos θ
T =q
k
[aT1 (M`≤1)P1
1 (cos θ)
+ aT2 (M`≤1)P12 (cos θ)
]
-1.0 -0.5 0.0 0.5 1.0
-1.0-0.5
0.00.51.01.52.0
G[ µb sr
]cos θ
G =q
k
[aG2 (M`≤1)P2
2 (cos θ)]
Y. Wunderlich Complete experiment in a TPWA
Effect of higher multipoles in lower (aL)αk
∆ resonance: ELABγ ' 340MeV → |M`>1|∣∣∣M∆
`≤1
∣∣∣ = |M`>1||M1+| < |E2−|
|M1+| ' 0.04
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
6
T[ µb sr
]
cos θ
T =q
k
[aT1 (M`≤2)P1
1 (cos θ)
+ aT2 (M`≤2)P12 (cos θ)
]
-1.0 -0.5 0.0 0.5 1.0
-1.0-0.5
0.00.51.01.52.0
G[ µb sr
]cos θ
G =q
k
[aG2 (M`≤2)P2
2 (cos θ)]
Y. Wunderlich Complete experiment in a TPWA
Effect of higher multipoles in lower (aL)αk
∆ resonance: ELABγ ' 340MeV → |M`>1|∣∣∣M∆
`≤1
∣∣∣ = |M`>1||M1+| < |E2−|
|M1+| ' 0.04
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
6
T[ µb sr
]
cos θ
T =q
k
[aT1 (M`≤2)P1
1 (cos θ)
+ aT2 (M`≤2)P12 (cos θ)
+ aT3 (M`≤2)P13 (cos θ)
]
-1.0 -0.5 0.0 0.5 1.0-1.5-1.0-0.5
0.00.51.01.52.0
G[ µb sr
]cos θ
G =q
k
[aG2 (M`≤2)P2
2 (cos θ)
+ aG3 (M`≤2)P23 (cos θ)
]Y. Wunderlich Complete experiment in a TPWA
Effect of higher multipoles in lower (aL)αk
∆ resonance: ELABγ ' 340MeV → |M`>1|∣∣∣M∆
`≤1
∣∣∣ = |M`>1||M1+| < |E2−|
|M1+| ' 0.04
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
6
T[ µb sr
]
cos θ
T =q
k
[aT1 (M`≤2)P1
1 + aT2 (M`≤2)P12
+ aT3 (M`≤2)P13 + aT4 (M`≤2)P1
4
]
-1.0 -0.5 0.0 0.5 1.0-1.5-1.0-0.5
0.00.51.01.52.0
G[ µb sr
]cos θ
G =q
k
[aG2 (M`≤2)P2
2 + aG3 (M`≤2)P23
+ aG4 (M`≤2)P24
]Y. Wunderlich Complete experiment in a TPWA
Effect of higher multipoles in lower (aL)αk
∆ resonance: ELABγ ' 340MeV → |M`>1|∣∣∣M∆
`≤1
∣∣∣ = |M`>1||M1+| < |E2−|
|M1+| ' 0.04
-1.0 -0.5 0.0 0.5 1.0-4
-2
0
2
4
6
T[ µb sr
]
cos θ
T =q
k
[aT1 (M`≤2)P1
1 + aT2 (M`≤2)P12
+ aT3 (M`≤2)P13 + aT4 (M`≤2)P1
4
]
-1.0 -0.5 0.0 0.5 1.0
-1.0-0.5
0.00.51.01.52.0
G[ µb sr
]cos θ
G =q
k
[aG2 (M`≤3)P2
2 + aG3 (M`≤3)P23
+ aG4 (M`≤3)P24
]Y. Wunderlich Complete experiment in a TPWA
Interference terms in bilinear forms of the (aL)αk
(aL)αk =[M∗`≤`max
M∗`>`max
]
(CL)αk(CL)αk
[(CL)αk
]† (CL)αk
M`≤`max
M`>`max
Hi
Y. Wunderlich Complete experiment in a TPWA
Interference terms in bilinear forms of the (aL)αk
(aL)αk =[M∗`≤`max
M∗`>`max
]
(CL)αk(CL)αk
[(CL)αk
]† (CL)αk
M`≤`max
M`>`max
Hi
=∑m,n
(M`≤)∗m [(CL)αk ]m,n (M`≤)n +∑p,q
(M`≤)∗p
[(CL)αk
]p,q
(M`>)q
+∑r ,l
(M`>)∗r
[(CL)αk
]†r ,l
(M`≤)l +∑b,c
(M`>)∗b
[(CL)αk
]b,c
(M`>)c
Y. Wunderlich Complete experiment in a TPWA
Interference terms in bilinear forms of the (aL)αk
(aL)αk =[M∗`≤`max
M∗`>`max
]
(CL)αk(CL)αk
[(CL)αk
]† (CL)αk
M`≤`max
M`>`max
Hi
=∑m,n
(M`≤)∗m [(CL)αk ]m,n (M`≤)n +∑p,q
(M`≤)∗p
[(CL)αk
]p,q
(M`>)q
+∑r ,l
(M`>)∗r
[(CL)αk
]†r ,l
(M`≤)l +∑b,c
(M`>)∗b
[(CL)αk
]b,c
(M`>)c
'∑m,n
(M`≤)∗m [(CL)αk ]m,n (M`≤)n + 2Re
[∑p,q
(M`≤)∗p
[(CL)αk
]p,q
(M`>)q
]︸ ︷︷ ︸
Interference term, causing trouble
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data, using `max > 1 multipoles
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3 are fixed parameters in lower (aL)αk ):
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data, using `max > 1 multipoles
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3 are fixed parameters in lower (aL)αk ):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data, using `max > 1 multipoles
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3 are fixed parameters in lower (aL)αk ):
Re[ E
0+
] [10−
3mπ
+]
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data, using `max > 1 multipoles
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3 are fixed parameters in lower (aL)αk ):
200 300 400 500 600 700 8000
2
4
6
8
Re[ E
0+
] [10−
3mπ
+]
200 300 400 500 600 700 800
0
5
10
15
20
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
200 300 400 500 600 700 800
-1.0
-0.5
0.0
Re[ E
1+
]
200 300 400 500 600 700 800-20-15-10
-505
Im[ M
1+
]
200 300 400 500 600 700 8000.0
0.5
1.0
1.5
2.0
2.5
Im[ M
1−]
200 300 400 500 600 700 800-1.0
-0.5
0.0
0.5
Im[ E
1+
]200 300 400 500 600 700 800
0
1
2
3
4
Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Fits to full MAID theory data, using `max > 1 multipoles
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3 are fixed parameters in lower (aL)αk ):
200 300 400 500 600 700 8000
2
4
6
8
Re[ E
0+
] [10−
3mπ
+]
200 300 400 500 600 700 800
0
5
10
15
20
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
200 300 400 500 600 700 800
-1.0
-0.5
0.0
Re[ E
1+
]
200 300 400 500 600 700 800-20-15-10
-505
Im[ M
1+
]
200 300 400 500 600 700 800-0.5
0.00.51.01.52.02.5
Im[ M
1−]
200 300 400 500 600 700 800-1.0
-0.5
0.0
0.5
Im[ E
1+
]200 300 400 500 600 700 800
0
1
2
3
4
Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Original MAID solutionY. Wunderlich Complete experiment in a TPWA
Conclusions and Outlook
I. Model independent TPWA fit approach developed using the totalcross section σ(W ) and the MATHEMATICA routineFindMinimum
II. Statements of [A. S. Omelaenko (1981)] verified by fitting theory datastemming from MAID2007 multipoles truncated at some order`max ≤ 4 (no contributions from higher multipoles)
→ Results confirmed and consistent up to `max = 4
III. Fit method also applicable for fitting the full MAID 2007 solution
→ Contributions of the higher multipoles M`>`max to the lower angular fitparameters aαk had to be taken into account in order to obtain sensiblesolutions
Y. Wunderlich Complete experiment in a TPWA
Conclusions and Outlook
I. Model independent TPWA fit approach developed using the totalcross section σ(W ) and the MATHEMATICA routineFindMinimum
II. Statements of [A. S. Omelaenko (1981)] verified by fitting theory datastemming from MAID2007 multipoles truncated at some order`max ≤ 4 (no contributions from higher multipoles)→ Results confirmed and consistent up to `max = 4
III. Fit method also applicable for fitting the full MAID 2007 solution→ Contributions of the higher multipoles M`>`max to the lower angular fit
parameters aαk had to be taken into account in order to obtain sensiblesolutions
Goal: Fitting real data
Study of the influence of statistical (and later systematic) errors(bootstrapping method)
Questions: Best model/parametrization for higher multipoles?, Howto fit the charged nπ+ channel?, ...
Y. Wunderlich Complete experiment in a TPWA
Appendices: Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,E ,G ,H,F generated using full MAID2007 (Fit `max = 1, NMC = 1000):
Y. Wunderlich Complete experiment in a TPWA
Appendices: Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,E ,G ,H,F generated using full MAID2007 (Fit `max = 1, NMC = 1000):
Re[ E
0+
] /mπ
+
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
Re[ E
1+
]Im[ M
1+
]Im[ M
1−]
Im[ E
1+
]Eγ [MeV]
Re[ M
1−]
Y. Wunderlich Complete experiment in a TPWA
Appendices: Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,E ,G ,H,F generated using full MAID2007 (Fit `max = 1, NMC = 1000):
200 300 400 500 600 700 8000
1
2
3
4
5
Re[ E
0+
] /mπ
+
200 300 400 500 600 700 800
-505
101520
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
200 300 400 500 600 700 800
-1.0
-0.5
0.0
0.5
Re[ E
1+
]
200 300 400 500 600 700 800
-20-15-10
-505
Im[ M
1+
]
200 300 400 500 600 700 8000
1
2
3
4
Im[ M
1−]
200 300 400 500 600 700 800
-1.0
-0.5
0.0
0.5
Im[ E
1+
]
200 300 400 500 600 700 800-2
-1
0
1
2
3
Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Appendices: Fits to full MAID theory data
Fit of observables σ0,Σ,T ,P,E ,G ,H,F generated using full MAID2007 (Fit `max = 1, NMC = 1000):
200 300 400 500 600 700 80001234567
Re[ E
0+
] /mπ
+
200 300 400 500 600 700 800
-505
101520
Eγ [MeV]
Re[ M
1+
]
Constraint:Im[E0+
]= 0 & Re
[E0+
]> 0
200300400500600700800
-1.0
-0.5
0.0
0.5
Re[ E
1+
]
200 300 400 500 600 700 800
-20-15-10
-505
Im[ M
1+
]
200 300 400 500 600 700 800
0
1
2
3
4
Im[ M
1−]
200300400500600700800
-1.0
-0.5
0.0
0.5
Im[ E
1+
]
200 300 400 500 600 700 800-2
-1
0
1
2
3
Eγ [MeV]
Re[ M
1−]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Original MAID solution
Y. Wunderlich Complete experiment in a TPWA
Appendices: Can higher multipoles fix the overall phase?
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3), without phase constraint:
Y. Wunderlich Complete experiment in a TPWA
Appendices: Can higher multipoles fix the overall phase?
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3), without phase constraint:
Re
[E0
+][
10−
3mπ
+]
Eγ [MeV]
Im[E
1+
]Re
[M1−
]
Im[E
0+
]Re
[M1
+]
Im[M
1−
]
Re
[E1
+]
Eγ [MeV]
Im[M
1+
]
Y. Wunderlich Complete experiment in a TPWA
Appendices: Can higher multipoles fix the overall phase?
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3), without phase constraint:
Re
[E0
+][
10−
3mπ
+]
Eγ [MeV]
Im[E
1+
]Re
[M1−
]
Im[E
0+
]Re
[M1
+]
Im[M
1−
]
Re
[E1
+]
Eγ [MeV]
Im[M
1+
]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Appendices: Can higher multipoles fix the overall phase?
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3), without phase constraint:
200 300 400 500 600 700 800-6
-4
-2
0
2
Re
[E0
+][
10−
3mπ
+]
200 300 400 500 600 700 800
-1.0
-0.5
0.0
0.5
Eγ [MeV]
Im[E
1+
]
200 300 400 500 600 700 800
-3
-2
-1
0
1
Re
[M1−
]
200 300 400 500 600 700 800
0
2
4
6
Im[E
0+
]
200 300 400 500 600 700 800-10
-5
0
5
10
15
Re
[M1
+]
200 300 400 500 600 700 8000.00.51.01.52.02.53.0
Im[M
1−
]200 300 400 500 600 700 800
-0.5
0.0
0.5
1.0
Re
[E1
+]
200 300 400 500 600 700 8000
5
10
15
20
25
Eγ [MeV]
Im[M
1+
]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Y. Wunderlich Complete experiment in a TPWA
Appendices: Can higher multipoles fix the overall phase?
Fit σ0,Σ,T ,P,G from full MAID 2007 (Fit `max = 1, NMC = 1000,higher multipoles 1 < `max ≤ 3), without phase constraint:
200 300 400 500 600 700 800-6
-4
-2
0
2
Re
[E0
+][
10−
3mπ
+]
200 300 400 500 600 700 800
-1.0
-0.5
0.0
0.5
Eγ [MeV]
Im[E
1+
]
200 300 400 500 600 700 800
-3
-2
-1
0
1
Re
[M1−
]
200 300 400 500 600 700 800
0
2
4
6
Im[E
0+
]
200 300 400 500 600 700 800-10
-5
0
5
10
15
Re
[M1
+]
200 300 400 500 600 700 8000.00.51.01.52.02.53.0
Im[M
1−
]200 300 400 500 600 700 800
-0.5
0.0
0.5
1.0
Re
[E1
+]
200 300 400 500 600 700 8000
5
10
15
20
25
Eγ [MeV]
Im[M
1+
]
Best χ2 solution:χ2Amb/χ
2Best ≈ 1012
Original MAID solution
Y. Wunderlich Complete experiment in a TPWA