[PPT] - Complete Experiments in pseudoscalar meson photoproduction Yannick PowerPoint Presentation

SLIDE 1

Complete Experiments in pseudoscalar meson photoproduction

Yannick Wunderlich

HISKP, University of Bonn

October 5, 2016

SLIDE 2

Motivation for photoproduction

]

2

Mass [GeV/c 1 2 3

p

J

+ 2 1 + 2 3 + 2 5 + 2 7 + 2 9 + 2 11

2

1

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N∗ spectrum
Predictions of the

Bonn CQM on the left [Loring et al. (2001)]

Resonances from

[PDG (2014)] on the right

[Andrew Wilson]

∗) Photoproduction is studied at photon facilities in Bonn (CBELSA/TAPS), Mainz (MAMI/A2), Newport News (CLAS), ...

Y. Wunderlich

Complete Experiments

SLIDE 3

Motivation for photoproduction

]

2

Mass [GeV/c 1 2 3

p

J

+ 2 1 + 2 3 + 2 5 + 2 7 + 2 9 + 2 11

2

1

2

3

2

5

2

7

2

9

2

11

****

939

****

1440

****

1520

****

1535

****

1650

****

1675

****

1680

***

1700

***

1710

****

1720

**

1860

***

1875

**

1880

**

1895

***

1900

**

1990

**

2000

*

2040

**

2060

*

2100

**

2120

****

2190

****

2220

****

2250

**

2300

**

2570

***

2600

N∗ spectrum
Predictions of the

Bonn CQM on the left [Loring et al. (2001)]

Resonances from

[PDG (2014)] on the right

[Andrew Wilson]

∗) Photoproduction is studied at photon facilities in Bonn (CBELSA/TAPS), Mainz (MAMI/A2), Newport News (CLAS), ... ∗) What I do: Identify a sufficient data base to get a unique solution for photoproduction amplitudes (or multipoles) → Complete Experiments.

Y. Wunderlich

Complete Experiments

SLIDE 4

Photoproduction amplitudes

Photoproduction amplitude in the CMS:

N γ B ϕ

= Tfi = Cχ†

msf

i

σ · ˆ ǫF1 + σ · ˆ q σ ·

ˆ

k × ˆ ǫ

F2 + i

σ · ˆ kˆ q · ˆ ǫF3 +i σ · ˆ qˆ q · ˆ ǫF4

χmsi

[Chew, Goldberger, Low & Nambu (1957)]

→ Process fully described by 4 complex amplitudes Fi (W , θ).

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Complete Experiments

SLIDE 5

Photoproduction amplitudes

Photoproduction amplitude in the CMS:

N γ B ϕ

= Tfi = Cχ†

msf

i

σ · ˆ ǫF1 + σ · ˆ q σ ·

ˆ

k × ˆ ǫ

F2 + i

σ · ˆ kˆ q · ˆ ǫF3 +i σ · ˆ qˆ q · ˆ ǫF4

χmsi

[Chew, Goldberger, Low & Nambu (1957)]

→ Process fully described by 4 complex amplitudes Fi (W , θ). Important concept: expansion of full amplitudes into partial waves:

F1 (W , θ) =

∞

ℓ=0
[ℓMℓ+ + Eℓ+] P

′

ℓ+1 (cos (θ)) + [(ℓ + 1) Mℓ− + Eℓ−] P

′

ℓ−1 (cos (θ))

F2 (W , θ) = . . .

N γ B ϕ JP; (I) ℓ

∗) J = |ℓ ± 1/2|, P = (−)ℓ+1. ∗) s-chn. resonance JP; (I)

multipole E (I)

ℓ±, M(I) ℓ±

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Complete Experiments

SLIDE 6

Photoproduction amplitudes

Photoproduction amplitude in the CMS:

N γ B ϕ

= Tfi = Cχ†

msf

i

σ · ˆ ǫF1 + σ · ˆ q σ ·

ˆ

k × ˆ ǫ

F2 + i

σ · ˆ kˆ q · ˆ ǫF3 +i σ · ˆ qˆ q · ˆ ǫF4

χmsi

[Chew, Goldberger, Low & Nambu (1957)]

→ Process fully described by 4 complex amplitudes Fi (W , θ). Important concept: expansion of full amplitudes into partial waves:

F1 (W , θ) =

∞ℓmax

ℓ=0
[ℓMℓ+ + Eℓ+] P

′

ℓ+1 (cos (θ)) + [(ℓ + 1) Mℓ− + Eℓ−] P

′

ℓ−1 (cos (θ))

F2 (W , θ) = . . .

N γ B ϕ JP; (I) ℓ

In practice: Truncate at some finite ℓmax → Try to extract the 4ℓmax complex multipoles in a fit to the data.

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Complete Experiments

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Polarization observables

Generic definition of an observable

Ω = β

σ0

dσ

dΩ

(B1,T1,R1) − dσ

dΩ

(B2,T2,R2)

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Complete Experiments

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Polarization observables

Generic definition of an observable

Ω = β

σ0

dσ

dΩ

(B1,T1,R1) − dσ

dΩ

(B2,T2,R2)

∗) In total, 16 non-redundant observables

Ωα (W , θ) = 1 2σ0

i,j

F ∗

i ˆ

Aα

ij Fj,

α = 1, . . . , 16

can be defined, involving Beam-, Target- and Recoil Polarization.

Beam Target Recoil Target + Recoil

x′

y′ z′ x′ x′ z′ z′

x

y z

x

z x z unpolarized

dσ

dΩ

0 = σ0

T P Tx′ Lx′ Tz′ Lz′ linear Σ H P G Ox′ T Oz′ circular F E Cx′ Cz′

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Observables in the transversity basis

Observable Transversity representation Type σ0

1 2

|b1|2 + |b2|2 + |b3|2 + |b4|2

ˇ Σ

1 2

− |b1|2 − |b2|2 + |b3|2 + |b4|2

S ˇ T

1 2

|b1|2 − |b2|2 − |b3|2 + |b4|2

ˇ P

1 2

− |b1|2 + |b2|2 − |b3|2 + |b4|2

ˇ G Im

−b1b∗

3 − b2b∗ 4

ˇ

H −Re

b1b∗

3 − b2b∗ 4

BT

ˇ E −Re

b1b∗

3 + b2b∗ 4

ˇ

F Im

b1b∗

3 − b2b∗ 4

ˇ

Ox′ −Re

−b1b∗

4 + b2b∗ 3

ˇ

Oz′ Im

−b1b∗

4 − b2b∗ 3

BR

ˇ Cx′ Im

b1b∗

4 − b2b∗ 3

ˇ

Cz′ Re

b1b∗

4 + b2b∗ 3

ˇ

Tx′ −Re

−b1b∗

2 + b3b∗ 4

ˇ

Tz′ −Im

b1b∗

2 − b3b∗ 4

T R

ˇ Lx′ −Im

−b1b∗

2 − b3b∗ 4

ˇ

Lz′ Re

−b1b∗

2 − b3b∗ 4

∗) Transversity amplitudes:

bi =

j MijFj.

∗) Different scheme of spin-quantization: msf | T |msi

tf | T |ti.

ti(tf ) = ± 1

2:

spin-projection of initial (final) baryon on the normal of the reaction plane. ∗) Observables simplify: ˇ Ωα = 1

2

i,j b∗

i ˜

Γα

ij bj.

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Complete Experiments

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Complete Experiments

∗) Question: How many and which observables ˇ Ωα have to be measured in order to uniquely extract the full amplitudes (e.g. transversity amplitudes bi)?.

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Complete Experiments

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Complete Experiments

∗) Mathematical solution: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Utilize b.t.p.-form ˇ Ωα = 1

2

i,j b∗

i ˜

Γα

ij bj and the completeness of the

˜ Γα-matrices (˜ Γα form an orthonormal basis): 1

4

α ˜

Γα

ba˜

Γα

st = δasδbt

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Complete Experiments

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Complete Experiments

∗) Mathematical solution: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Utilize b.t.p.-form ˇ Ωα = 1

2

i,j b∗

i ˜

Γα

ij bj and the completeness of the

˜ Γα-matrices (˜ Γα form an orthonormal basis): 1

4

α ˜

Γα

ba˜

Γα

st = δasδbt

α

˜ Γα

ac ˇ

Ωα = 1 2

α

˜ Γα

ac

i,j

b∗

i ˜

Γα

ij bj = 1

2

i,j

b∗

i

α

˜ Γα

ac˜

Γα

ij bj

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Complete Experiments

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Complete Experiments

∗) Mathematical solution: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Utilize b.t.p.-form ˇ Ωα = 1

2

i,j b∗

i ˜

Γα

ij bj and the completeness of the

˜ Γα-matrices (˜ Γα form an orthonormal basis): 1

4

α ˜

Γα

ba˜

Γα

st = δasδbt

α

˜ Γα

ac ˇ

Ωα = 1 2

α

˜ Γα

ac

i,j

b∗

i ˜

Γα

ij bj = 1

2

i,j

b∗

i

α

˜ Γα

ac˜

Γα

ij bj

= 2

i,j

b∗

i δicδajbj = 2b∗ cba

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Complete Experiments

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Complete Experiments

∗) Mathematical solution: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Utilize b.t.p.-form ˇ Ωα = 1

2

i,j b∗

i ˜

Γα

ij bj and the completeness of the

˜ Γα-matrices (˜ Γα form an orthonormal basis): 1

4

α ˜

Γα

ba˜

Γα

st = δasδbt

α

˜ Γα

ac ˇ

Ωα = 1 2

α

˜ Γα

ac

i,j

b∗

i ˜

Γα

ij bj = 1

2

i,j

b∗

i

α

˜ Γα

ac˜

Γα

ij bj

= 2

i,j

b∗

i δicδajbj = 2b∗ cba

→ b∗

i bj = 1

2

α
˜

Γα

ij

∗ ˇ Ωα → |bi| =

b∗

i bi & eφij =

b∗

j bi

|bj| |bi|

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Complete Experiments

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Complete Experiments

∗) Mathematical solution: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Utilize b.t.p.-form ˇ Ωα = 1

2

i,j b∗

i ˜

Γα

ij bj and the completeness of the

˜ Γα-matrices (˜ Γα form an orthonormal basis): 1

4

α ˜

Γα

ba˜

Γα

st = δasδbt

b∗

i bj = 1

2

α
˜

Γα

ij

∗ ˇ Ωα → |bi| =

b∗

i bi & eφij =

b∗

j bi

|bj| |bi|

Re Im ˜ b1 ˜ b2 ˜ b3 ˜ b4 φ21 φ32 φ43 Re Im φ (W , θ) b1 b2 b3 b4 φ21 φ32 φ43

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Complete Experiments

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Complete Experiments

∗) Mathematical solution: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Utilize b.t.p.-form ˇ Ωα = 1

2

i,j b∗

i ˜

Γα

ij bj and the completeness of the

˜ Γα-matrices (˜ Γα form an orthonormal basis): 1

4

α ˜

Γα

ba˜

Γα

st = δasδbt

b∗

i bj = 1

2

α
˜

Γα

ij

∗ ˇ Ωα → |bi| =

b∗

i bi & eφij =

b∗

j bi

|bj| |bi| ∗) Use “Fierz-identities” ˇ Ωα ˇ Ωβ = Cαβ

δη ˇ

Ωδ ˇ Ωη (with known coefficients Cαβ

δη ) to prove:

8 observables can yield |bi| & φij.
Double-polarization obs. with

recoil-polarization (type BR and T R) have to be measured.

No more than two observables from the

same double-polarization class are allowed.

The phase φ(W , θ) remains

undetermined.

Re Im ˜ b1 ˜ b2 ˜ b3 ˜ b4 φ21 φ32 φ43 Re Im φ (W , θ) b1 b2 b3 b4 φ21 φ32 φ43

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Complete Experiments

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Complete Experiments

∗) Full amplitudes: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] ˇ Ωα = 1

2 b| ˜

Γα |b ↔ b∗

i bj = 1 2

α
˜

Γα

ij

∗ ˇ Ωα → 8 observables

Re Im ˜ b1 ˜ b2 ˜ b3 ˜ b4 φ21 φ32 φ43 Re Im φ (W , θ) b1 b2 b3 b4 φ21 φ32 φ43

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Complete Experiments

SLIDE 18

Complete Experiments

∗) Full amplitudes: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] ˇ Ωα = 1

2 b| ˜

Γα |b ↔ b∗

i bj = 1 2

α
˜

Γα

ij

∗ ˇ Ωα → 8 observables

Re Im ˜ b1 ˜ b2 ˜ b3 ˜ b4 φ21 φ32 φ43 Re Im φ (W , θ) b1 b2 b3 b4 φ21 φ32 φ43

∗) Multipoles: consider TPWA truncated at ℓmax

ˇ Ωα (W , θ) = ρ

2ℓmax+βα+γα

k=βα

(aL)

ˇ Ωα k

(W ) Pβα

k

(cos θ) , (aL)

ˇ Ωα k

(W ) = Mℓmax (W )| (CL)

ˇ Ωα k

|Mℓmax (W ) ,

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Complete Experiments

SLIDE 19

Complete Experiments

∗) Full amplitudes: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] ˇ Ωα = 1

2 b| ˜

Γα |b ↔ b∗

i bj = 1 2

α
˜

Γα

ij

∗ ˇ Ωα → 8 observables

Re Im ˜ b1 ˜ b2 ˜ b3 ˜ b4 φ21 φ32 φ43 Re Im φ (W , θ) b1 b2 b3 b4 φ21 φ32 φ43

∗) Multipoles: consider TPWA truncated at ℓmax

ˇ Ωα (W , θ) = ρ

2ℓmax+βα+γα

k=βα

(aL)

ˇ Ωα k

(W ) Pβα

k

(cos θ) , (aL)

ˇ Ωα k

(W ) = Mℓmax (W )| (CL)

ˇ Ωα k

|Mℓmax (W ) ,

→ Algebraic inversion of this bilinear equation system not possible

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Complete Experiments

SLIDE 20

Complete Experiments

∗) Full amplitudes: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] ˇ Ωα = 1

2 b| ˜

Γα |b ↔ b∗

i bj = 1 2

α
˜

Γα

ij

∗ ˇ Ωα → 8 observables

Re Im ˜ b1 ˜ b2 ˜ b3 ˜ b4 φ21 φ32 φ43 Re Im φ (W , θ) b1 b2 b3 b4 φ21 φ32 φ43

∗) Multipoles: consider TPWA truncated at ℓmax

ˇ Ωα (W , θ) = ρ

2ℓmax+βα+γα

k=βα

(aL)

ˇ Ωα k

(W ) Pβα

k

(cos θ) , (aL)

ˇ Ωα k

(W ) = Mℓmax (W )| (CL)

ˇ Ωα k

|Mℓmax (W ) ,

→ Algebraic inversion of this bilinear equation system not possible → Instead: [Omelaenko, (1981)] & [Y.W., R. Beck and L. Tiator, (2014)] Study discrete ambiguities of the group S {σ0, Σ, T, P} = ⇒ ”Exact“ TPWA can be complete with just 5 observables, e.g. {σ0, Σ, T, P, F}.

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Complete Experiments

SLIDE 21

Details on the multipole Fit procedure

Two step method:

1. Fit the angular distributions of observables, parametrized by

ˇ Ωα (W , θ) = ρ

2ℓmax+βα+γα

k=βα

(aL)α

k (W ) Pβα k

(cos θ)

⇒ Angular fit parameters

aFit

L

α

k

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SLIDE 22

Details on the multipole Fit procedure

Two step method:

1. Fit the angular distributions of observables, parametrized by

ˇ Ωα (W , θ) = ρ

2ℓmax+βα+γα

k=βα

(aL)α

k (W ) Pβα k

(cos θ)

⇒ Angular fit parameters

aFit

L

α

k

2. Minimize the function (”multi-indices” (i, j) = ({α, k} , {α′, k′})):

χ2 =

i,j

aFit

L

i − Mℓ| (CL)i |Mℓ
C−1

ij

aFit

L

j − Mℓ| (CL)j |Mℓ
,

using the MATHEMATICA method

FindMinimum

χ2 (Mℓ) , {{Re [E0+] , (x1)0} , . . . , {Im [Mℓmax−] , (yn)0}}
and varying the real and imaginary parts of the (possibly phase

constrained) multipoles in the fit. Cij is the covariance matrix of the Legendre coefficients stemming from step 1.

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Generation of start values for FindMinimum

1. The total cross section

ˆ σ (W ) = ℓmax

Mℓ cMℓ |Mℓ|2

constrains the (8ℓmax − 1)-dimensional multipole space Mℓ. Re[E0+] Mℓ \ Re[E0+]

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Generation of start values for FindMinimum

1. The total cross section

ˆ σ (W ) = ℓmax

Mℓ cMℓ |Mℓ|2

constrains the (8ℓmax − 1)-dimensional multipole space Mℓ.

2. ˆ

σ (W ) defines an (8ℓmax − 2)- dimensional ellipsoid in Mℓ. Re[E0+] Mℓ \ Re[E0+]

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Complete Experiments

SLIDE 25

Generation of start values for FindMinimum

1. The total cross section

ˆ σ (W ) = ℓmax

Mℓ cMℓ |Mℓ|2

constrains the (8ℓmax − 1)-dimensional multipole space Mℓ.

2. ˆ

σ (W ) defines an (8ℓmax − 2)- dimensional ellipsoid in Mℓ.

3. Solutions to the TPWA problem lie on

the ellipsoid defined by ˆ σ (W ). Re[E0+] Mℓ \ Re[E0+]

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Complete Experiments

SLIDE 26

Generation of start values for FindMinimum

1. The total cross section

ˆ σ (W ) = ℓmax

Mℓ cMℓ |Mℓ|2

constrains the (8ℓmax − 1)-dimensional multipole space Mℓ.

2. ˆ

σ (W ) defines an (8ℓmax − 2)- dimensional ellipsoid in Mℓ.

3. Solutions to the TPWA problem lie on

the ellipsoid defined by ˆ σ (W ).

4. The start values for the

FindMinimum-Fit are chosen randomly on the ˆ σ (W )-ellipsoid. ⇒ Monte Carlo sampling of the multipole space. Re[E0+] Mℓ \ Re[E0+]

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Generation of start values for FindMinimum

5. A FindMinimum-minimization is

performed for each of the randomly generated start configurations. ⇒ NMC = # of M.C. start configurations = # of (possibly redundant) solutions Re[E0+] Mℓ \ Re[E0+]

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Complete Experiments

SLIDE 28

Generation of start values for FindMinimum

5. A FindMinimum-minimization is

performed for each of the randomly generated start configurations. ⇒ NMC = # of M.C. start configurations = # of (possibly redundant) solutions

6. Analysis described up to now is fully

model-independent. However: if wished for or needed, individual partial-wave parameters can be fixed to model-constraints quite freely. Re[E0+] Mℓ \ Re[E0+]

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SLIDE 29

Generation of start values for FindMinimum

6. Analysis described up to now is fully

model-independent. However: if wished for or needed, individual partial-wave parameters can be fixed to model-constraints quite freely.

7. In this way, map out the global

minimum as well as all local minima

f the χ2-function.

Re[E0+] Mℓ \ Re[E0+]

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Description of the fitted datasets

The following datasets were investigated for γp → π0p:

I. Data taken at the MAMI facility:
σ0: 266 energy points for E LAB

γ

∈ [218, 1573] MeV

[P. Adlarson et al., arXiv:1506.08849 [hep-ex]]

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SLIDE 31

Description of the fitted datasets

The following datasets were investigated for γp → π0p:

I. Data taken at the MAMI facility:
σ0: 266 energy points for E LAB

γ

∈ [218, 1573] MeV

[P. Adlarson et al., arXiv:1506.08849 [hep-ex]]

II. Data taken at the GRAAL facility:
Σ: 31 energy points for E LAB

γ

∈ [551, 1450] MeV

[O. Bartalini et al., Eur. Phys. J. A 26, 399 (2005)]

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Complete Experiments

SLIDE 32

Description of the fitted datasets

The following datasets were investigated for γp → π0p:

I. Data taken at the MAMI facility:
σ0: 266 energy points for E LAB

γ

∈ [218, 1573] MeV

[P. Adlarson et al., arXiv:1506.08849 [hep-ex]]

II. Data taken at the GRAAL facility:
Σ: 31 energy points for E LAB

γ

∈ [551, 1450] MeV

[O. Bartalini et al., Eur. Phys. J. A 26, 399 (2005)]

III. Data from CBELSA/TAPS:
T: 24 energy points for E LAB

γ

∈ [700, 1900] MeV

P: 8 (!) energy points, i.e. E LAB

γ

∈ [650, 950] MeV

H: 8 (!) energy points, i.e. E LAB

γ

∈ [650, 950] MeV for all 3 obs. cf. [J. Hartmann et al., Phys. Lett. B 748 (2015), prelim.]

E: 33 energy points for E LAB

γ

∈ [600, 2300] MeV

[M. Gottschall et al., Phys. Rev. Lett. 112 no. 1, 012003 (2014)]

G: 19 energy points for E LAB

γ

∈ [630, 1950] MeV

[A. Thiel et al., Phys. Rev. Lett. 109, 102001 (2012)]

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Complete Experiments

SLIDE 33

Description of the fitted datasets

The following datasets were investigated for γp → π0p:

I. Data taken at the MAMI facility:
σ0: 266 energy points for E LAB

γ

∈ [218, 1573] MeV

[P. Adlarson et al., arXiv:1506.08849 [hep-ex]]

II. Data taken at the GRAAL facility:
Σ: 31 energy points for E LAB

γ

∈ [551, 1450] MeV

[O. Bartalini et al., Eur. Phys. J. A 26, 399 (2005)]

III. Data from CBELSA/TAPS:
T: 24 energy points for E LAB

γ

∈ [700, 1900] MeV

P: 8 (!) energy points, i.e. E LAB

γ

∈ [650, 950] MeV

H: 8 (!) energy points, i.e. E LAB

γ

∈ [650, 950] MeV for all 3 obs. cf. [J. Hartmann et al., Phys. Lett. B 748 (2015), prelim.]

E: 33 energy points for E LAB

γ

∈ [600, 2300] MeV

[M. Gottschall et al., Phys. Rev. Lett. 112 no. 1, 012003 (2014)]

G: 19 energy points for E LAB

γ

∈ [630, 1950] MeV

[A. Thiel et al., Phys. Rev. Lett. 109, 102001 (2012)]

→ Datasets overlap on 8 (!) energy-points E LAB

γ

∈ [650, 950] MeV!

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SLIDE 34

Description of the fitted datasets

The following datasets were investigated for γp → π0p: {σ0, Σ, T, P, E, G, H}. From investigations of the angular distributions of the data (and later confirmed by χ2/ndf in the multipole fit): ℓmax = 2 and ℓmax = 3 truncation approximations can already describe the data.

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SLIDE 35

Results of fully un-constrained analyses

ℓmax = 2-fit (NMC = 16000)

I. χ2-plot:
There exists a global

minimum.

ℓmax = 3-fit (NMC = 24000)

I. χ2-plot:
There exists a global

minimum.

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Complete Experiments

SLIDE 36

Results of fully un-constrained analyses

ℓmax = 2-fit (NMC = 16000)

I. χ2-plot:
There exists a global

minimum.

Global min. is well separated

from other local minima.

ℓmax = 3-fit (NMC = 24000)

I. χ2-plot:
There exists a global

minimum.

Local minima (ambiguities)

exists that have a very similar χ2 to the global min.

Y. Wunderlich

Complete Experiments

SLIDE 37

Results of fully un-constrained analyses

ℓmax = 2-fit (NMC = 16000)

I. blank
II. Example multipole:

: BnGa 2014-02 : ˆ

σ(W )-interval

ℓmax = 3-fit (NMC = 24000)

I. blank
II. Example multipole:
Y. Wunderlich

Complete Experiments

SLIDE 38

Results of fully un-constrained analyses

ℓmax = 2-fit (NMC = 16000)

I. blank
II. Example multipole:

χ2

best/ndf + 0.5

: BnGa 2014-02

: ˆ

σ(W )-interval

ℓmax = 3-fit (NMC = 24000)

I. blank
II. Example multipole:

χ2

best/ndf + 0.05

Y. Wunderlich

Complete Experiments

SLIDE 39

Results of fully un-constrained analyses

ℓmax = 2-fit (NMC = 16000)

I. blank
II. Example multipole:

χ2

best/ndf + 1.0

: BnGa 2014-02

: ˆ

σ(W )-interval

ℓmax = 3-fit (NMC = 24000)

I. blank
II. Example multipole:

χ2

best/ndf + 0.3

Y. Wunderlich

Complete Experiments

SLIDE 40

Results of fully un-constrained analyses

ℓmax = 2-fit (NMC = 16000)

I. blank
II. blank
III. Issues:
Under-fitting for higher Eγ
Missing the S, F-, P, F-

and D, F-interferences.

ℓmax = 3-fit (NMC = 24000)

I. blank
II. blank
III. Issues:
Over-fitting for lower Eγ
Accidential ambiguities:

NAC

max = 1 2

42×3 − 1
= 2047.
Y. Wunderlich

Complete Experiments

SLIDE 41

Results of fully un-constrained analyses

ℓmax = 2-fit (NMC = 16000)

I. blank
II. blank
III. Issues:
Under-fitting for higher Eγ
Missing the S, F-, P, F-

and D, F-interferences.

ℓmax = 3-fit (NMC = 24000)

I. blank
II. blank
III. Issues:
Over-fitting for lower Eγ
Accidential ambiguities:

NAC

max = 1 2

42×3 − 1
= 2047.

→ Way out: Introduce model dependence by fixing the F-waves to BnGa 2014-02, letting all other mutlipoles float freely in the fit.

Y. Wunderlich

Complete Experiments

SLIDE 42

χ2

ndf vs. Eγ for the fit including BnGa-F-waves

Y. Wunderlich

Complete Experiments

SLIDE 43

χ2

ndf vs. Eγ for the fit including BnGa-F-waves

Y. Wunderlich

Complete Experiments

SLIDE 44

The best solution for S-, P- and D-waves

: BnGa 2014-02 : ˆ

σ(W )-interval

Y. Wunderlich

Complete Experiments

SLIDE 45

The best solution for S-, P- and D-waves

: BnGa 2014-02 : ˆ

σ(W )-interval

Y. Wunderlich

Complete Experiments

SLIDE 46

S-, P- and D-waves in the interval

χ2

best + 1.0

: BnGa 2014-02

: ˆ

σ(W )-interval

Y. Wunderlich

Complete Experiments

SLIDE 47

S-, P- and D-waves in the interval

χ2

best + 1.0

: BnGa 2014-02

: ˆ

σ(W )-interval

Y. Wunderlich

Complete Experiments

SLIDE 48

Results for the S-, P- and D-waves

700 750 800 850 900 6 8 10 12 EΓMeV ReE0

Cm Fm

700 750 800 850 900 0.4 0.2 0.0 0.2 0.4 0.6 EΓMeV ReE1

Cm Fm

700 750 800 850 900 0.8 0.6 0.4 0.2 0.0 0.2 EΓMeV Im E1

Cm Fm

700 750 800 850 900 0.0 0.5 1.0 1.5 2.0 EΓMeV ReM1

Cm Fm

700 750 800 850 900 1 2 3 4 5 EΓMeV Im M1

Cm Fm

700 750 800 850 900 1 2 3 4 5 6 7 8 EΓMeV ReM1

Cm Fm

700 750 800 850 900 1 1 2 EΓMeV Im M1

Cm Fm

Errors: bootstrapping

: BnGa 2014-02 : MAID-07 : SAID CM12

Y. Wunderlich

Complete Experiments

SLIDE 49

Results for the S-, P- and D-waves

700 750 800 850 900 0.05 0.10 0.15 0.20 0.25 0.30 0.35 EΓMeV ReE2

Cm Fm

700 750 800 850 900 0.4 0.3 0.2 0.1 0.0 0.1 EΓMeV Im E2

Cm Fm

700 750 800 850 900 1 2 3 4 5 6 EΓMeV ReE2

Cm Fm

700 750 800 850 900 2 2 4 6 EΓMeV Im E2

Cm Fm

700 750 800 850 900 0.3 0.2 0.1 0.0 0.1 0.2 EΓMeV ReM2

Cm Fm

700 750 800 850 900 0.6 0.4 0.2 0.0 0.2 EΓMeV Im M2

Cm Fm

700 750 800 850 900 0.5 1.0 1.5 2.0 2.5 3.0 3.5 EΓMeV ReM2

Cm Fm

700 750 800 850 900 2.5 2.0 1.5 1.0 0.5 0.0 0.5 EΓMeV Im M2

Cm Fm

Errors: bootstrapping

: BnGa 2014-02 : MAID-07 : SAID CM12

Y. Wunderlich

Complete Experiments

SLIDE 50

Summary & Outlook

∗) A monte-carlo sampling fit method was applied to {σ0, Σ, T, P, E, G, H}-data for γp → π0p in the 2nd resonance region.

Y. Wunderlich

Complete Experiments

SLIDE 51

Summary & Outlook

∗) A monte-carlo sampling fit method was applied to {σ0, Σ, T, P, E, G, H}-data for γp → π0p in the 2nd resonance region.

→ ”LFits“ suggest an ℓmax = 2 (or 3)-truncation to describe the data. → ℓmax = 2 multipole fit: the best solution is ”unique“ but χ2 too large (high-low partial wave interferences!) → ℓmax = 3 multipole fit: a ”unique“ global minimum exists, however there are many side-minima (ambiguities!) → S-, P-wave multipoles varied, F-waves fixed to BnGa: Monte Carlo method yields a global minimum, well separated from other local minima. χ2/ndf and the behaviour of the solution are reasonable.

Y. Wunderlich

Complete Experiments

SLIDE 52

Summary & Outlook

∗) A monte-carlo sampling fit method was applied to {σ0, Σ, T, P, E, G, H}-data for γp → π0p in the 2nd resonance region.

→ ”LFits“ suggest an ℓmax = 2 (or 3)-truncation to describe the data. → ℓmax = 2 multipole fit: the best solution is ”unique“ but χ2 too large (high-low partial wave interferences!) → ℓmax = 3 multipole fit: a ”unique“ global minimum exists, however there are many side-minima (ambiguities!) → S-, P-wave multipoles varied, F-waves fixed to BnGa: Monte Carlo method yields a global minimum, well separated from other local minima. χ2/ndf and the behaviour of the solution are reasonable.

∗) What to do with the obtained solution?

→ Fitting a model independent pole+background parametrization (”L+P“-method of Alfred ˇ Svarc): yields D13-pole-parameters → Iteration of multipole-fitting with BnGa-code applied to SE-results: under construction ...

Y. Wunderlich

Complete Experiments

SLIDE 53

Thank You!

SLIDE 54

Additional Slides

SLIDE 55

Details on the multipole fit procedure II

Ansatz: Use the total cross section ˆ σ (W ). Example: ℓ ≤ ℓmax = 1, phase constraint Im

E C

0+

= 0 & Re
E C

0+

> 0:

ˆ σ (W ) ≈ 4π q

k

Re
E C

0+

2 + 6 Re

E C

1+

2 + 6 Im

E C

1+

2 + 2 Re

MC

1+

2 +2 Im

MC

1+

2 + Re

MC

1−

2 + Im

MC

1−

2

Y. Wunderlich

Complete Experiments

SLIDE 56

Details on the multipole fit procedure II

Ansatz: Use the total cross section ˆ σ (W ). Example: ℓ ≤ ℓmax = 1, phase constraint Im

E C

0+

= 0 & Re
E C

0+

> 0:

ˆ σ (W ) ≈ 4π q

k

Re
E C

0+

2 + 6 Re

E C

1+

2 + 6 Im

E C

1+

2 + 2 Re

MC

1+

2 +2 Im

MC

1+

2 + Re

MC

1−

2 + Im

MC

1−

2 ∗) ˆ σ (W ) constrains the intervals of the multipoles:

Re

E C

0+

∈
0,
k

q ˆ σ(W ) 4π

, . . ., Im
MC

1−

∈
−
k

q ˆ σ(W ) 4π ,

k

q ˆ σ(W ) 4π

∗) The total cross section, being quadratic form in the multipoles, also

defines an ellipsoid in the multipole space.

Y. Wunderlich

Complete Experiments

SLIDE 57

Cut selections for solution “data”

Cut on solutions χ2

j with χ2

j −χ2 best

ndf

< ǫ Mathematical ambiguity χ2 Mℓ Unique best solution χ2 Mℓ The χ2 is defined by the fitted Legendre coefficients

aFit

L

α

k .

Y. Wunderlich

Complete Experiments

SLIDE 58

Cut selections for solution “data”

Cut on solutions χ2

j with χ2

j −χ2 best

ndf

< ǫ Mathematical ambiguity χ2 Mℓ Unique best solution χ2 Mℓ Start values have been distributed on the relevant part of the space Mℓ.

Y. Wunderlich

Complete Experiments

SLIDE 59

Cut selections for solution “data”

Cut on solutions χ2

j with χ2

j −χ2 best

ndf

< ǫ Mathematical ambiguity χ2 Mℓ Unique best solution χ2 Mℓ Minimizations of χ2 converge within several iterations.

Y. Wunderlich

Complete Experiments

SLIDE 60

Cut selections for solution “data”

Cut on solutions χ2

j with χ2

j −χ2 best

ndf

< ǫ Mathematical ambiguity χ2 Mℓ Unique best solution χ2 Mℓ Minimizations of χ2 converge within several iterations.

Y. Wunderlich

Complete Experiments

SLIDE 61

Cut selections for solution “data”

Cut on solutions χ2

j with χ2

j −χ2 best

ndf

< ǫ Mathematical ambiguity χ2 Mℓ Unique best solution χ2 Mℓ Minimizations of χ2 converge within several iterations.

Y. Wunderlich

Complete Experiments

SLIDE 62

Cut selections for solution “data”

Cut on solutions χ2

j with χ2

j −χ2 best

ndf

< ǫ Mathematical ambiguity χ2 Mℓ Unique best solution χ2 Mℓ Minimizations of χ2 converge within several iterations.

Y. Wunderlich

Complete Experiments

SLIDE 63

Cut selections for solution “data”

Cut on solutions χ2

j with χ2

j −χ2 best

ndf

< ǫ Mathematical ambiguity χ2 Mℓ Unique best solution χ2 Mℓ Cut selection using ǫ = 1

Y. Wunderlich

Complete Experiments

SLIDE 64

Cut selections for solution “data”

Cut on solutions χ2

j with χ2

j −χ2 best

ndf

< ǫ Mathematical ambiguity χ2 Mℓ Unique best solution χ2 Mℓ Cut selection using ǫ = 1 / Cut selection using ǫ ∼ num.precision

Y. Wunderlich

Complete Experiments

SLIDE 65

Bootstrapping

∗) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)]: Estimate an unknown distribution function of a random variable R

X1, . . . , Xn, ˆ

F

, by

generating bootstrap random samples xb = (x∗

1, . . . , x∗ n) from the data

(x1, . . . , xn) and approximating the R-distribution-fct. by Rb

xb, ˆ

F

.
Y. Wunderlich

Complete Experiments

SLIDE 66

Bootstrapping

∗) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)]: Estimate an unknown distribution function of a random variable R

X1, . . . , Xn, ˆ

F

, by

generating bootstrap random samples xb = (x∗

1, . . . , x∗ n) from the data

(x1, . . . , xn) and approximating the R-distribution-fct. by Rb

xb, ˆ

F

.
Y. Wunderlich

Complete Experiments

SLIDE 67

Bootstrapping

∗) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)]: Estimate an unknown distribution function of a random variable R

X1, . . . , Xn, ˆ

F

, by

generating bootstrap random samples xb = (x∗

1, . . . , x∗ n) from the data

(x1, . . . , xn) and approximating the R-distribution-fct. by Rb

xb, ˆ

F

.
Y. Wunderlich

Complete Experiments

SLIDE 68

Bootstrapping

∗) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)]: Estimate an unknown distribution function of a random variable R

X1, . . . , Xn, ˆ

F

, by

generating bootstrap random samples xb = (x∗

1, . . . , x∗ n) from the data

(x1, . . . , xn) and approximating the R-distribution-fct. by Rb

xb, ˆ

F

.
Use gaussian distribution

function centered at Ωα, with σ = ∆Ωα. → Resample data with replacement. → Ensemble of (1 + NEns.) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter.

Y. Wunderlich

Complete Experiments

SLIDE 69

Bootstrapping

∗) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)]: Estimate an unknown distribution function of a random variable R

X1, . . . , Xn, ˆ

F

, by

generating bootstrap random samples xb = (x∗

1, . . . , x∗ n) from the data

(x1, . . . , xn) and approximating the R-distribution-fct. by Rb

xb, ˆ

F

.
NEns. = 1
Use gaussian distribution

function centered at Ωα, with σ = ∆Ωα. → Resample data with replacement. → Ensemble of (1 + NEns.) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter.

Y. Wunderlich

Complete Experiments

SLIDE 70

Bootstrapping

∗) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)]: Estimate an unknown distribution function of a random variable R

X1, . . . , Xn, ˆ

F

, by

generating bootstrap random samples xb = (x∗

1, . . . , x∗ n) from the data

(x1, . . . , xn) and approximating the R-distribution-fct. by Rb

xb, ˆ

F

.
NEns. = 5
Use gaussian distribution

function centered at Ωα, with σ = ∆Ωα. → Resample data with replacement. → Ensemble of (1 + NEns.) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter.

Y. Wunderlich

Complete Experiments

SLIDE 71

Bootstrapping

∗) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)]: Estimate an unknown distribution function of a random variable R

X1, . . . , Xn, ˆ

F

, by

generating bootstrap random samples xb = (x∗

1, . . . , x∗ n) from the data

(x1, . . . , xn) and approximating the R-distribution-fct. by Rb

xb, ˆ

F

.
NEns. = 10
Use gaussian distribution

function centered at Ωα, with σ = ∆Ωα. → Resample data with replacement. → Ensemble of (1 + NEns.) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter.

Y. Wunderlich

Complete Experiments

SLIDE 72

Bootstrapping

∗) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)]: Estimate an unknown distribution function of a random variable R

X1, . . . , Xn, ˆ

F

, by

generating bootstrap random samples xb = (x∗

1, . . . , x∗ n) from the data

(x1, . . . , xn) and approximating the R-distribution-fct. by Rb

xb, ˆ

F

.
NEns. = 400
Use gaussian distribution

function centered at Ωα, with σ = ∆Ωα. → Resample data with replacement. → Ensemble of (1 + NEns.) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter.

Y. Wunderlich

Complete Experiments

SLIDE 73

Bootstrap histograms for Eγ = 883.8 MeV

6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 0.0 0.5 1.0 1.5 2.0 ReE0

Cm Fm

0.4 0.3 0.2 0.1 0.0 0.1 0.2 1 2 3 4 ReE1

Cm Fm

0.60 0.55 0.50 0.45 0.40 2 4 6 8 10 Im E1

Cm Fm

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1 2 3 4 ReM1

Cm Fm

0.6 0.4 0.2 0.0 1 2 3 4 Im M1

Cm Fm

3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 ReM1

Cm Fm

0.8 0.6 0.4 0.2 0.0 0.2 0.0 0.5 1.0 1.5 2.0 2.5 Im M1

Cm Fm

Y. Wunderlich

Complete Experiments

SLIDE 74

Bootstrap histograms for Eγ = 883.8 MeV

0.10 0.15 0.20 0.25 0.30 0.35 0.40 2 4 6 8 ReE2

Cm Fm

0.04 0.020.00 0.02 0.04 0.06 5 10 15 20 25 30 Im E2

Cm Fm

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ReE2

Cm Fm

4.304.354.404.454.504.554.604.65 2 4 6 8 Im E2

Cm Fm

0.15 0.10 0.050.00 0.05 0.10 0.15 2 4 6 8 10 ReM2

Cm Fm

0.20 0.15 0.10 0.050.00 0.05 0.10 2 4 6 8 Im M2

Cm Fm

1.7 1.8 1.9 2.0 2.1 2.2 2.3 1 2 3 4 ReM2

Cm Fm

1.1 1.0 0.9 0.8 0.7 0.6 1 2 3 4 5 Im M2

Cm Fm

Y. Wunderlich

Complete Experiments

SLIDE 75

Bootstrap histograms for Eγ = 883.8 MeV

6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 0.0 0.5 1.0 1.5 2.0 ReE0

Cm Fm

0.4 0.3 0.2 0.1 0.0 0.1 0.2 1 2 3 4 ReE1

Cm Fm

0.60 0.55 0.50 0.45 0.40 2 4 6 8 10 Im E1

Cm Fm

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1 2 3 4 ReM1

Cm Fm

0.6 0.4 0.2 0.0 1 2 3 4 Im M1

Cm Fm

3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 ReM1

Cm Fm

0.8 0.6 0.4 0.2 0.0 0.2 0.0 0.5 1.0 1.5 2.0 2.5 Im M1

Cm Fm

Y. Wunderlich

Complete Experiments

SLIDE 76

Bootstrap histograms for Eγ = 883.8 MeV

0.10 0.15 0.20 0.25 0.30 0.35 0.40 2 4 6 8 ReE2

Cm Fm

0.04 0.020.00 0.02 0.04 0.06 5 10 15 20 25 30 Im E2

Cm Fm

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ReE2

Cm Fm

4.304.354.404.454.504.554.604.65 2 4 6 8 Im E2

Cm Fm

0.15 0.10 0.050.00 0.05 0.10 0.15 2 4 6 8 10 ReM2

Cm Fm

0.20 0.15 0.10 0.050.00 0.05 0.10 2 4 6 8 Im M2

Cm Fm

1.7 1.8 1.9 2.0 2.1 2.2 2.3 1 2 3 4 ReM2

Cm Fm

1.1 1.0 0.9 0.8 0.7 0.6 1 2 3 4 5 Im M2

Cm Fm

Y. Wunderlich

Complete Experiments

SLIDE 77

Bootstrap histograms for Eγ = 883.8 MeV

6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 0.0 0.5 1.0 1.5 2.0 ReE0

Cm Fm

0.4 0.3 0.2 0.1 0.0 0.1 0.2 1 2 3 4 ReE1

Cm Fm

0.60 0.55 0.50 0.45 0.40 2 4 6 8 10 Im E1

Cm Fm

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1 2 3 4 ReM1

Cm Fm

0.6 0.4 0.2 0.0 1 2 3 4 Im M1

Cm Fm

3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 ReM1

Cm Fm

0.8 0.6 0.4 0.2 0.0 0.2 0.0 0.5 1.0 1.5 2.0 2.5 Im M1

Cm Fm

Y. Wunderlich

Complete Experiments

SLIDE 78

Bootstrap histograms for Eγ = 883.8 MeV

0.10 0.15 0.20 0.25 0.30 0.35 0.40 2 4 6 8 ReE2

Cm Fm

0.04 0.020.00 0.02 0.04 0.06 5 10 15 20 25 30 Im E2

Cm Fm

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ReE2

Cm Fm

4.304.354.404.454.504.554.604.65 2 4 6 8 Im E2

Cm Fm

0.15 0.10 0.050.00 0.05 0.10 0.15 2 4 6 8 10 ReM2

Cm Fm

0.20 0.15 0.10 0.050.00 0.05 0.10 2 4 6 8 Im M2

Cm Fm

1.7 1.8 1.9 2.0 2.1 2.2 2.3 1 2 3 4 ReM2

Cm Fm

1.1 1.0 0.9 0.8 0.7 0.6 1 2 3 4 5 Im M2

Cm Fm

Y. Wunderlich

Complete Experiments

SLIDE 79

Bootstrap histograms for Eγ = 883.8 MeV

6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 0.0 0.5 1.0 1.5 2.0 ReE0

Cm Fm

0.4 0.3 0.2 0.1 0.0 0.1 0.2 1 2 3 4 ReE1

Cm Fm

0.60 0.55 0.50 0.45 0.40 2 4 6 8 10 Im E1

Cm Fm

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1 2 3 4 ReM1

Cm Fm

0.6 0.4 0.2 0.0 1 2 3 4 Im M1

Cm Fm

3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 ReM1

Cm Fm

0.8 0.6 0.4 0.2 0.0 0.2 0.0 0.5 1.0 1.5 2.0 2.5 Im M1

Cm Fm

Y. Wunderlich

Complete Experiments

SLIDE 80

Bootstrap histograms for Eγ = 883.8 MeV

0.10 0.15 0.20 0.25 0.30 0.35 0.40 2 4 6 8 ReE2

Cm Fm

0.04 0.020.00 0.02 0.04 0.06 5 10 15 20 25 30 Im E2

Cm Fm

1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ReE2

Cm Fm

4.304.354.404.454.504.554.604.65 2 4 6 8 Im E2

Cm Fm

0.15 0.10 0.050.00 0.05 0.10 0.15 2 4 6 8 10 ReM2

Cm Fm

0.20 0.15 0.10 0.050.00 0.05 0.10 2 4 6 8 Im M2

Cm Fm

1.7 1.8 1.9 2.0 2.1 2.2 2.3 1 2 3 4 ReM2

Cm Fm

1.1 1.0 0.9 0.8 0.7 0.6 1 2 3 4 5 Im M2

Cm Fm

Y. Wunderlich

Complete Experiments

SLIDE 81

Polarization Observables - more detail

Problem: 4 complex amplitudes Fi (W , θ) ≡ 8 real numbers ⇒ 1 observable dσ

dΩ

0 insufficient to determine the amplitudes!
Y. Wunderlich

Complete Experiments

SLIDE 82

Polarization Observables - more detail

Problem: 4 complex amplitudes Fi (W , θ) ≡ 8 real numbers ⇒ 1 observable dσ

dΩ

0 insufficient to determine the amplitudes!

Solution: Utilize the polarization degrees of freedom of the reaction

Example: Beam- Polarization

φ

[A. Thiel, PhD (2012)]

⇒

N ∝ dσ

dΩ

φ [deg]

← reaction plane Eγ = 1000 MeV

Y. Wunderlich

Complete Experiments

SLIDE 83

Polarization Observables - more detail

Problem: 4 complex amplitudes Fi (W , θ) ≡ 8 real numbers ⇒ 1 observable dσ

dΩ

0 insufficient to determine the amplitudes!

Solution: Utilize the polarization degrees of freedom of the reaction

Example: Beam- Polarization

φ

[A. Thiel, PhD (2012)]

⇒

N ∝ dσ

dΩ

φ [deg]

← reaction plane Eγ = 1000 MeV

The observable Σ appears as amplitude of the φ-modulation dσ

dΩ

(θ, φ) =

dσ

dΩ

0 (1 − ǫLΣ cos(2φ)).

Σ is an asymmetry between different polarization states: Σ =

1 2( dσ

dΩ)0

dσ

dΩ

(⊥,0,0) − dσ

dΩ

(,0,0) .

Y. Wunderlich

Complete Experiments

SLIDE 84

Complete Experiment: truncated partial wave analysis

∗) Question: How many and which observables are needed if multipoles {Eℓ±, Mℓ±} are the goal in a PWA truncated at some ℓmax?.

Y. Wunderlich

Complete Experiments

SLIDE 85

Complete Experiment: truncated partial wave analysis

∗) Important hint: [A.S. Omelaenko, Sov. J. Nucl. Phys. 34, 406 (1981).] Study of discrete ambiguities in a TPWA using the following trick: switch cos(θ) ↔ t := tan (θ/2), use b4(θ) = b3(−θ), b2(θ) = b1(−θ) and do a linear factor decomposition of b2 and b4: b2 (θ) = −Ca2ℓmax exp

i θ

2

(1 + t2)ℓmax [(t − β1) (t − β2) . . . (t − β2ℓmax)]

b4 (θ) = Ca2ℓmax exp

i θ

2

(1 + t2)ℓmax [(t − α1) (t − α2) . . . (t − α2ℓmax)]

→ A set of Omelaenko-roots {αk, βk} is fully equivalent to a multipole-solution {Eℓ±, Mℓ±}.

Y. Wunderlich

Complete Experiments

SLIDE 86

Complete Experiment: truncated partial wave analysis

∗) Important hint: [A.S. Omelaenko, Sov. J. Nucl. Phys. 34, 406 (1981).] b2 (θ) = −Ca2ℓmax

exp(i θ

2)

(1+t2)ℓmax [(t − β1) (t − β2) . . . (t − β2ℓmax)]

b4 (θ) = Ca2ℓmax

exp(i θ

2)

(1+t2)ℓmax [(t − α1) (t − α2) . . . (t − α2ℓmax)]

∗) Complex conjugation of all {αk, βk}, or some subset of them, leaves |bi|2 invariant and therefore also the group S

σ0, ˇ

Σ, ˇ T, ˇ P

≡ ˇ

ΩαS = 1

2

± |b1|2 ± |b2|2 ± |b3|2 + |b4|2
Y. Wunderlich

Complete Experiments

SLIDE 87

Complete Experiment: truncated partial wave analysis

∗) Important hint: [A.S. Omelaenko, Sov. J. Nucl. Phys. 34, 406 (1981).] b2 (θ) = −Ca2ℓmax

exp(i θ

2)

(1+t2)ℓmax [(t − β1) (t − β2) . . . (t − β2ℓmax)]

b4 (θ) = Ca2ℓmax

exp(i θ

2)

(1+t2)ℓmax [(t − α1) (t − α2) . . . (t − α2ℓmax)]

∗) Complex conjugation of all {αk, βk}, or some subset of them, leaves |bi|2 invariant and therefore also the group S

σ0, ˇ

Σ, ˇ T, ˇ P

≡ ˇ

ΩαS = 1

2

± |b1|2 ± |b2|2 ± |b3|2 + |b4|2

∗) Omelaenko’s constraint

k

αk =

k′

βk′ has to be fulfilled.

Y. Wunderlich

Complete Experiments

SLIDE 88

Complete Experiment: truncated partial wave analysis

∗) Important hint: [A.S. Omelaenko, Sov. J. Nucl. Phys. 34, 406 (1981).] b2 (θ) = −Ca2ℓmax

exp(i θ

2)

(1+t2)ℓmax [(t − β1) (t − β2) . . . (t − β2ℓmax)]

b4 (θ) = Ca2ℓmax

exp(i θ

2)

(1+t2)ℓmax [(t − α1) (t − α2) . . . (t − α2ℓmax)]

∗) Complex conjugation of {αk, βk} leaves

σ0, ˇ

Σ, ˇ T, ˇ P

invariant.

∗) Omelaenko’s constraint

k αk = k′ βk′.

→ 2 kinds of symmetries/ambiguities: double ambiguity

αk → α∗

k, βk → β∗ k for all k

One has always

k αk = k′ βk′

→

k α∗ k = k′ β∗ k′

→ Mathematically exact symmetry

Is resolved by F, G, as well as any

BR and T R observable.

accidential ambiguities

Conjugation of subset of {αk, βk}

→

˜

αk, ˜ βk

Very (very) likely

k ˜

αk ≃

k′ ˜

βk′ → Manifest as approximate symmetry

Is resolved by in principle any
bservable.
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Complete Experiments

SLIDE 89

Which ℓmax to choose? → “LFit-method”

∗) Utilize the parametrization of the angular distributions of polarization

bservables ˇ

Ωα as expansions into Pm

ℓ (cos θ) for fixed energy:

ˇ Ωα (W , θ) =

2ℓmax+βα+γα

k=βα

(aL)α

k (W ) Pβα k

(cos θ) .

Y. Wunderlich

Complete Experiments

SLIDE 90

Which ℓmax to choose? → “LFit-method”

∗) Utilize the parametrization of the angular distributions of polarization

bservables ˇ

Ωα as expansions into Pm

ℓ (cos θ) for fixed energy:

ˇ Ωα (W , θ) =

2ℓmax+βα+γα

k=βα

(aL)α

k (W ) Pβα k

(cos θ) . → Fit angular distributions with some low initial ℓmax (ℓmax = 0 most commonly) and see if χ2/ndf is satifactory. If not: → Raise truncation order by 1 and do new fit until

χ2/ndf
≈ 1.

→ Hint for dominant partial wave by the order ℓmax at which this procedure terminates.

Y. Wunderlich

Complete Experiments

SLIDE 91

Which ℓmax to choose? → “LFit-method”

∗) Utilize the parametrization of the angular distributions of polarization

bservables ˇ

Ωα as expansions into Pm

ℓ (cos θ) for fixed energy:

ˇ Ωα (W , θ) =

2ℓmax+βα+γα

k=βα

(aL)α

k (W ) Pβα k

(cos θ) . → Fit angular distributions with some low initial ℓmax (ℓmax = 0 most commonly) and see if χ2/ndf is satifactory. If not: → Raise truncation order by 1 and do new fit until

χ2/ndf
≈ 1.

→ Hint for dominant partial wave by the order ℓmax at which this procedure terminates. ∗) Nice: Procedure is simple, model-independent and furthermore reliably reflects the capability of the data to give infomation on higher partial wave contributions.

Y. Wunderlich

Complete Experiments

SLIDE 92

LFits to {σ0, Σ, T, P, E, G, H}

To be published in [Y. W., F. Afzal, A. Thiel and R. Beck, (2016)]

W [MeV] 1200 1400 1600 1800 /ndf

2

χ 10 20 30 40

=1

max

L =2

max

L =3

max

L =4

max

L =5

max

L

[MeV]

γ

E 500 1000 1500

P r e l i m i n a r y

σ0

W [MeV] 1400 1600 1800 /ndf

2

χ 20 40 =1

max

L =2

max

L =3

max

L =4

max

L [MeV]

γ

E 600 800 1000 1200 1400

P r e l i m i n a r y

ˇ Σ

W [MeV] 1600 1800 2000 /ndf

2

χ 10 20 =1

max

L =2

max

L =3

max

L =4

max

L [MeV]

γ

E 1000 1500

P r e l i m i n a r y

ˇ T

W [MeV] 1500 1550 1600 /ndf

2

χ 2 4 6 =1

max

L =2

max

L =3

max

L =4

max

L [MeV]

γ

E 700 800 900

P r e l i m i n a r y

ˇ P

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Complete Experiments

SLIDE 93

LFits to {σ0, Σ, T, P, E, G, H}

To be published in [Y. W., F. Afzal, A. Thiel and R. Beck, (2016)]

W [MeV] 1600 1800 2000 2200 /ndf

2

χ 5 10 15 20 =1

max

L =2

max

L =3

max

L =4

max

L [MeV]

γ

E 1000 1500 2000

P r e l i m i n a r y

ˇ E

W [MeV] 1500 1600 1700 1800 /ndf

2

χ 5 10 15 20 =1

max

L =2

max

L =3

max

L =4

max

L [MeV]

γ

E 800 1000 1200

P r e l i m i n a r y

ˇ G

W [MeV] 1500 1550 1600 /ndf

2

χ 0.5 1 1.5 2 2.5 =1

max

L =2

max

L =3

max

L =4

max

L [MeV]

γ

E 700 800 900

P r e l i m i n a r y

ˇ H Overall, ℓmax = 2 should be OK in all energy bins E LAB

γ

∈ [650, 950] MeV except maybe the last 2 bins.

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Complete Experiments

SLIDE 94

χ2

best vs. Eγ for the ℓmax = 2-fit

Y. Wunderlich

Complete Experiments

SLIDE 95

χ2

best vs. Eγ for the ℓmax = 2-fit

Y. Wunderlich

Complete Experiments

SLIDE 96

The best solution for S-, P- and D-waves

Y. Wunderlich

Complete Experiments

SLIDE 97

The best solution for S-, P- and D-waves

Y. Wunderlich

Complete Experiments

SLIDE 98

S-, P- and D-waves in the interval

χ2

best + 0.5

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Complete Experiments

SLIDE 99

S-, P- and D-waves in the interval

χ2

best + 0.5

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Complete Experiments

SLIDE 100

S-, P- and D-waves in the interval

χ2

best + 1.0

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Complete Experiments

SLIDE 101

S-, P- and D-waves in the interval

χ2

best + 1.0

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Complete Experiments

SLIDE 102

S-, P- and D-waves in the interval

χ2

best + 4.0

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Complete Experiments

SLIDE 103

S-, P- and D-waves in the interval

χ2

best + 4.0

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Complete Experiments

SLIDE 104

Problems with the ℓmax = 2 multipole fit

There exists a unique, well-separated (in χ2) solution for ℓmax = 2, however: (i) χ2/ndf is too large for all energy bins except the first 2-4. (ii) Solution does not make sense compared to models (more precisely, to BnGa 2014-02).

Y. Wunderlich

Complete Experiments

SLIDE 105

Partial wave interferences in Legendre coefficients

(aL)α

k =

M∗

ℓ≤ℓmax

M∗

ℓ>ℓmax



        (CL)α

k

˜ CL α

k

˜ CL α

k

†

ˆ

CL α

k

                 Mℓ≤ℓmax Mℓ>ℓmax Hi        

∗) In the (aL)α

k , partial waves with ℓmax ≥ 3 may interfere with those

having ℓmax ≤ 3 but the LFits may only hint at this, or not show this at all!

Y. Wunderlich

Complete Experiments

SLIDE 106

Partial wave interferences in Legendre coefficients

(aL)α

k =

M∗

ℓ≤ℓmax

M∗

ℓ>ℓmax



        (CL)α

k

˜ CL α

k

˜ CL α

k

†

ˆ

CL α

k

                 Mℓ≤ℓmax Mℓ>ℓmax Hi        

∗) In the (aL)α

k , partial waves with ℓmax ≥ 3 may interfere with those

having ℓmax ≤ 3 but the LFits may only hint at this, or not show this at all! ∗) In case the multipole fit has all partial waves Mℓ with ℓ ≥ 3 set equal to zero, it has no chance to take into account the interferences and modify the results for S-, P-, and D-waves accordingly.

Y. Wunderlich

Complete Experiments

SLIDE 107

Partial wave interferences in Legendre coefficients

(aL)α

k =

M∗

ℓ≤ℓmax

M∗

ℓ>ℓmax



        (CL)α

k

˜ CL α

k

˜ CL α

k

†

ˆ

CL α

k

                 Mℓ≤ℓmax Mℓ>ℓmax Hi        

∗) In the (aL)α

k , partial waves with ℓmax ≥ 3 may interfere with those

having ℓmax ≤ 3 but the LFits may only hint at this, or not show this at all! ∗) In case the multipole fit has all partial waves Mℓ with ℓ ≥ 3 set equal to zero, it has no chance to take into account the interferences and modify the results for S-, P-, and D-waves accordingly. → One has to at least take into account F-waves into the fitting in some way! → Fit a truncation at ℓmax = 3 and let the F-waves run freely in the fit.

Y. Wunderlich

Complete Experiments

SLIDE 108

χ2

best vs. Eγ for the ℓmax = 3-fit

Y. Wunderlich

Complete Experiments

SLIDE 109

χ2

best vs. Eγ for the ℓmax = 3-fit

Y. Wunderlich

Complete Experiments

SLIDE 110

The best solution for S-, P-, D- and F-waves

Y. Wunderlich

Complete Experiments

SLIDE 111

The best solution for S-, P-, D- and F-waves

Y. Wunderlich

Complete Experiments

SLIDE 112

The best solution for S-, P-, D- and F-waves

Y. Wunderlich

Complete Experiments

SLIDE 113

S-, P-, D- and F-waves in the interval

χ2

best + 0.05

Y. Wunderlich

Complete Experiments

SLIDE 114

S-, P-, D- and F-waves in the interval

χ2

best + 0.05

Y. Wunderlich

Complete Experiments

SLIDE 115

S-, P-, D- and F-waves in the interval

χ2

best + 0.05

Y. Wunderlich

Complete Experiments

SLIDE 116

S-, P-, D- and F-waves in the interval

χ2

best + 0.1

Y. Wunderlich

Complete Experiments

SLIDE 117

S-, P-, D- and F-waves in the interval

χ2

best + 0.1

Y. Wunderlich

Complete Experiments

SLIDE 118

S-, P-, D- and F-waves in the interval

χ2

best + 0.1

Y. Wunderlich

Complete Experiments

SLIDE 119

S-, P-, D- and F-waves in the interval

χ2

best + 0.2

Y. Wunderlich

Complete Experiments

SLIDE 120

S-, P-, D- and F-waves in the interval

χ2

best + 0.2

Y. Wunderlich

Complete Experiments

SLIDE 121

S-, P-, D- and F-waves in the interval

χ2

best + 0.2

Y. Wunderlich

Complete Experiments

SLIDE 122

Problems with the ℓmax = 3 multipole fit

There exists a global minimum, which is however not well separated from the other local minima of χ2!

Y. Wunderlich

Complete Experiments

SLIDE 123

Problems with the ℓmax = 3 multipole fit

There exists a global minimum, which is however not well separated from the other local minima of χ2! Reasons: (i) Equation set defined by

aFit

L

α

k is not ”compatible“ (≡ exactly

solvable).

Y. Wunderlich

Complete Experiments

SLIDE 124

Problems with the ℓmax = 3 multipole fit

There exists a global minimum, which is however not well separated from the other local minima of χ2! Reasons: (i) Equation set defined by

aFit

L

α

k is not ”compatible“ (≡ exactly

solvable). (ii) There exist

1 2

42ℓmax − 2
= 2047

candidates for accidential ambiguities for ℓmax = 3! Way out: fix F-waves to a model, here: BnGa 2014-02.

Y. Wunderlich

Complete Experiments

SLIDE 125

Bootstrap results for the S-, P- and D-waves - Whole plot interval

700 750 800 850 900 5 10 15 EΓMeV ReE0

Cm Fm

700 750 800 850 900 6 4 2 2 4 6 EΓMeV ReE1

Cm Fm

700 750 800 850 900 6 4 2 2 4 6 EΓMeV Im E1

Cm Fm

700 750 800 850 900 10 5 5 10 EΓMeV ReM1

Cm Fm

700 750 800 850 900 10 5 5 10 EΓMeV Im M1

Cm Fm

700 750 800 850 900 15 10 5 5 10 15 EΓMeV ReM1

Cm Fm

700 750 800 850 900 15 10 5 5 10 15 EΓMeV Im M1

Cm Fm

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Complete Experiments

SLIDE 126

Bootstrap results for the S-, P- and D-waves - Whole plot interval

700 750 800 850 900 4 2 2 4 EΓMeV ReE2

Cm Fm

700 750 800 850 900 4 2 2 4 EΓMeV Im E2

Cm Fm

700 750 800 850 900 10 5 5 10 EΓMeV ReE2

Cm Fm

700 750 800 850 900 10 5 5 10 EΓMeV Im E2

Cm Fm

700 750 800 850 900 4 2 2 4 EΓMeV ReM2

Cm Fm

700 750 800 850 900 4 2 2 4 EΓMeV Im M2

Cm Fm

700 750 800 850 900 6 4 2 2 4 6 EΓMeV ReM2

Cm Fm

700 750 800 850 900 6 4 2 2 4 6 EΓMeV Im M2

Cm Fm

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Complete Experiments