The complete experiment problem of pseudoscalar meson ... · Yannick Wunderlich HISKP, University...

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 )?


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


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?


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]



Two step method:



k=βα

(aL)αk (W )Pβα

k (cos θ)


)αk


k (cos θ)

- Pβα


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.



Two step method:



k=βα

(aL)αk (W )Pβα

k (cos θ)


)αk


k (cos θ)

- Pβα


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.




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 )




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.


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+]




2. σ(W ) defines an (8`max − 2)-dimensional ellipsoid in M`.

Re[E

0+

]

M` \ Re[E0+]





3. Solutions to the TPWA problem lie onthe ellipsoid defined by σ(W ).

Re[E

0+

]

M` \ Re[E0+]





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+]



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+]



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+]


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−]




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




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


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−]




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




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−]


2Best ≈ 1012




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−]


2Best ≈ 1012

Original MAID solution


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−]




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−]


2Best ≈ 1012




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−]


2Best ≈ 1012




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−]


2Best ≈ 1012



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




`≤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)]




`≤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)]




`≤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



`≤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




`≤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


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∗`>`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∗`>`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


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−]




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−]


2Best ≈ 1012




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−]


2Best ≈ 1012




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−]


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


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?, ...


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−]




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−]


2Best ≈ 1012




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−]


2Best ≈ 1012



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+

]




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+

]


2Best ≈ 1012




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+

]


2Best ≈ 1012


The complete experiment problem of pseudoscalar meson ... · Yannick Wunderlich HISKP, University...

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