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Complete Experiments in pseudoscalar meson photoproduction Yannick Wunderlich HISKP, University of Bonn October 5, 2016 Motivation for photoproduction 3 - N spectrum *** - Predictions of the ** 2600 2570 Bonn CQM on the ** ****

  1. Complete Experiments ∗ ) Mathematical solution: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Ω α = 1 Utilize b.t.p.-form ˇ i ˜ i , j b ∗ Γ α � ij b j and the completeness of the 2 Γ α form an orthonormal basis): 1 Γ α -matrices (˜ ˜ α ˜ ba ˜ Γ α Γ α � st = δ as δ bt 4 b ∗ � ∗ ˇ j b i i b j = 1 � Ω α → | b i | = i b i & e φ ij = b ∗ � ˜ Γ α � b ∗ ij 2 | b j | | b i | α Ω α ˇ Ω β = C αβ Ω δ ˇ ∗ ) Use “Fierz-identities” ˇ δη ˇ Ω η Im ˜ b 2 (with known coefficients C αβ ˜ δη ) to prove: b 3 φ 32 φ 21 - 8 observables can yield | b i | & φ ij . ˜ φ 43 b 4 - Double-polarization obs. with ˜ b 1 recoil-polarization (type BR and T R ) Re Im have to be measured. b 1 - No more than two observables from the φ 21 b 2 same double-polarization class are φ ( W , θ ) φ 32 allowed. b 3 - The phase φ ( W , θ ) remains φ 43 Re undetermined. b 4 Y. Wunderlich Complete Experiments

  2. Complete Experiments ∗ ) Full amplitudes: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Im Im Ω α = 1 Γ α | b � ↔ ˇ 2 � b | ˜ ˜ b 2 b 1 ˜ b 2 φ 21 b 3 � ∗ ˇ φ 32 � φ ( W , θ ) φ 21 i b j = 1 ˜ b ∗ Γ α Ω α � φ 32 ˜ φ 43 2 α ij b 4 ˜ b 1 b 3 → 8 observables φ 43 Re Re b 4 Y. Wunderlich Complete Experiments

  3. Complete Experiments ∗ ) Full amplitudes: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Im Im Ω α = 1 Γ α | b � ↔ ˇ 2 � b | ˜ ˜ b 2 b 1 ˜ b 2 φ 21 b 3 � ∗ ˇ φ 32 � φ ( W , θ ) φ 21 i b j = 1 ˜ b ∗ Γ α Ω α � φ 32 ˜ φ 43 2 α ij b 4 ˜ b 1 b 3 → 8 observables φ 43 Re Re b 4 ∗ ) Multipoles: consider TPWA truncated at ℓ max 2 ℓ max + β α + γ α Ω α ( W , θ ) = ρ ˇ ˇ � Ω α ( W ) P β α ( a L ) (cos θ ) , k k k = β α ˇ ˇ Ω α Ω α ( a L ) ( W ) = �M ℓ max ( W ) | ( C L ) |M ℓ max ( W ) � , k k Y. Wunderlich Complete Experiments

  4. Complete Experiments ∗ ) Full amplitudes: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Im Im Ω α = 1 Γ α | b � ↔ ˇ 2 � b | ˜ ˜ b 2 b 1 ˜ b 2 φ 21 b 3 � ∗ ˇ φ 32 � φ ( W , θ ) φ 21 i b j = 1 ˜ b ∗ Γ α Ω α � φ 32 ˜ φ 43 2 α ij b 4 ˜ b 1 b 3 → 8 observables φ 43 Re Re b 4 ∗ ) Multipoles: consider TPWA truncated at ℓ max 2 ℓ max + β α + γ α Ω α ( W , θ ) = ρ ˇ ˇ � Ω α ( W ) P β α ( a L ) (cos θ ) , k k k = β α ˇ ˇ Ω α Ω α ( a L ) ( W ) = �M ℓ max ( W ) | ( C L ) |M ℓ max ( W ) � , k k → Algebraic inversion of this bilinear equation system not possible Y. Wunderlich Complete Experiments

  5. Complete Experiments ∗ ) Full amplitudes: [Chiang & Tabakin, Phys. Rev. C 55, 2054 (1997)] Im Im Ω α = 1 Γ α | b � ↔ ˇ 2 � b | ˜ ˜ b 2 b 1 ˜ b 2 φ 21 b 3 � ∗ ˇ φ 32 � φ ( W , θ ) φ 21 i b j = 1 ˜ b ∗ Γ α Ω α � φ 32 ˜ φ 43 2 α ij b 4 ˜ b 1 b 3 → 8 observables φ 43 Re Re b 4 ∗ ) Multipoles: consider TPWA truncated at ℓ max 2 ℓ max + β α + γ α ˇ Ω α ( W , θ ) = ρ ˇ � Ω α ( W ) P β α ( a L ) (cos θ ) , k k k = β α ˇ ˇ Ω α Ω α ( a L ) ( W ) = �M ℓ max ( W ) | ( C L ) |M ℓ max ( W ) � , k k → 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 { σ 0 , Σ , T , P } = ⇒ ”Exact“ TPWA can be complete with just 5 observables, e.g. { σ 0 , Σ , T , P , F } . Y. Wunderlich Complete Experiments

  6. Details on the multipole Fit procedure Two step method: 1. Fit the angular distributions of observables, parametrized by 2 ℓ max + β α + γ α Ω α ( W , θ ) = ρ ˇ ( a L ) α k ( W ) P β α � (cos θ ) k k = β α � α ⇒ Angular fit parameters � a Fit L k Y. Wunderlich Complete Experiments

  7. Details on the multipole Fit procedure Two step method: 1. Fit the angular distributions of observables, parametrized by 2 ℓ max + β α + γ α Ω α ( W , θ ) = ρ ˇ ( a L ) α k ( W ) P β α � (cos θ ) k k = β α � α ⇒ Angular fit parameters � a Fit L k 2. Minimize the function (”multi-indices” ( i , j ) = ( { α, k } , { α ′ , k ′ } )): χ 2 = � � � � � � � C − 1 a Fit � a Fit � i − �M ℓ | ( C L ) i |M ℓ � j − �M ℓ | ( C L ) j |M ℓ � , i , j L ij L using the MATHEMATICA method χ 2 ( M ℓ ) , {{ Re [ E 0+ ] , ( x 1 ) 0 } , . . . , { Im [ M ℓ max − ] , ( y n ) 0 }} � � FindMinimum and varying the real and imaginary parts of the (possibly phase constrained) multipoles in the fit. C ij is the covariance matrix of the Legendre coefficients stemming from step 1. Y. Wunderlich Complete Experiments

  8. Generation of start values for FindMinimum 1. The total cross section σ ( W ) = � ℓ max M ℓ c M ℓ |M ℓ | 2 ˆ constrains the (8 ℓ max − 1)-dimensional multipole space M ℓ . Re [ E 0+ ] M ℓ \ Re [ E 0+ ] Y. Wunderlich Complete Experiments

  9. Generation of start values for FindMinimum 1. The total cross section σ ( W ) = � ℓ max M ℓ c M ℓ |M ℓ | 2 ˆ constrains the (8 ℓ max − 1)-dimensional multipole space M ℓ . 2. ˆ σ ( W ) defines an (8 ℓ max − 2)- Re [ E 0+ ] dimensional ellipsoid in M ℓ . M ℓ \ Re [ E 0+ ] Y. Wunderlich Complete Experiments

  10. Generation of start values for FindMinimum 1. The total cross section σ ( W ) = � ℓ max M ℓ c M ℓ |M ℓ | 2 ˆ constrains the (8 ℓ max − 1)-dimensional multipole space M ℓ . 2. ˆ σ ( W ) defines an (8 ℓ max − 2)- Re [ E 0+ ] dimensional ellipsoid in M ℓ . 3. Solutions to the TPWA problem lie on the ellipsoid defined by ˆ σ ( W ). M ℓ \ Re [ E 0+ ] Y. Wunderlich Complete Experiments

  11. Generation of start values for FindMinimum 1. The total cross section σ ( W ) = � ℓ max M ℓ c M ℓ |M ℓ | 2 ˆ constrains the (8 ℓ max − 1)-dimensional multipole space M ℓ . 2. ˆ σ ( W ) defines an (8 ℓ max − 2)- Re [ E 0+ ] dimensional ellipsoid in M ℓ . 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. M ℓ \ Re [ E 0+ ] Y. Wunderlich Complete Experiments

  12. Generation of start values for FindMinimum 5. A FindMinimum -minimization is performed for each of the randomly generated start configurations. ⇒ N MC = # of M.C. start configurations Re [ E 0+ ] = # of (possibly redundant) solutions M ℓ \ Re [ E 0+ ] Y. Wunderlich Complete Experiments

  13. Generation of start values for FindMinimum 5. A FindMinimum -minimization is performed for each of the randomly generated start configurations. ⇒ N MC = # of M.C. start configurations Re [ E 0+ ] = # 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. M ℓ \ Re [ E 0+ ] Y. Wunderlich Complete Experiments

  14. 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. Re [ E 0+ ] 7. In this way, map out the global minimum as well as all local minima of the χ 2 -function. M ℓ \ Re [ E 0+ ] Y. Wunderlich Complete Experiments

  15. Description of the fitted datasets The following datasets were investigated for γ p → π 0 p : I. Data taken at the MAMI facility: - σ 0 : 266 energy points for E LAB ∈ [218 , 1573] MeV γ [P. Adlarson et al., arXiv:1506.08849 [hep-ex]] Y. Wunderlich Complete Experiments

  16. Description of the fitted datasets The following datasets were investigated for γ p → π 0 p : I. Data taken at the MAMI facility: - σ 0 : 266 energy points for E LAB ∈ [218 , 1573] MeV γ [P. Adlarson et al., arXiv:1506.08849 [hep-ex]] II. Data taken at the GRAAL facility: - Σ: 31 energy points for E LAB ∈ [551 , 1450] MeV γ [O. Bartalini et al., Eur. Phys. J. A 26, 399 (2005)] Y. Wunderlich Complete Experiments

  17. Description of the fitted datasets The following datasets were investigated for γ p → π 0 p : I. Data taken at the MAMI facility: - σ 0 : 266 energy points for E LAB ∈ [218 , 1573] MeV γ [P. Adlarson et al., arXiv:1506.08849 [hep-ex]] II. Data taken at the GRAAL facility: - Σ: 31 energy points for E LAB ∈ [551 , 1450] MeV γ [O. Bartalini et al., Eur. Phys. J. A 26, 399 (2005)] III. Data from CBELSA/TAPS: - T : 24 energy points for E LAB ∈ [700 , 1900] MeV γ - P : 8 (!) energy points, i.e. E LAB ∈ [650 , 950] MeV γ - H : 8 (!) energy points, i.e. E LAB ∈ [650 , 950] MeV γ for all 3 obs. cf. [J. Hartmann et al., Phys. Lett. B 748 (2015), prelim.] - E : 33 energy points for E LAB ∈ [600 , 2300] MeV γ [M. Gottschall et al., Phys. Rev. Lett. 112 no. 1, 012003 (2014)] - G : 19 energy points for E LAB ∈ [630 , 1950] MeV γ [A. Thiel et al., Phys. Rev. Lett. 109, 102001 (2012)] Y. Wunderlich Complete Experiments

  18. Description of the fitted datasets The following datasets were investigated for γ p → π 0 p : I. Data taken at the MAMI facility: - σ 0 : 266 energy points for E LAB ∈ [218 , 1573] MeV γ [P. Adlarson et al., arXiv:1506.08849 [hep-ex]] II. Data taken at the GRAAL facility: - Σ: 31 energy points for E LAB ∈ [551 , 1450] MeV γ [O. Bartalini et al., Eur. Phys. J. A 26, 399 (2005)] III. Data from CBELSA/TAPS: - T : 24 energy points for E LAB ∈ [700 , 1900] MeV γ - P : 8 (!) energy points, i.e. E LAB ∈ [650 , 950] MeV γ - H : 8 (!) energy points, i.e. E LAB ∈ [650 , 950] MeV γ for all 3 obs. cf. [J. Hartmann et al., Phys. Lett. B 748 (2015), prelim.] - E : 33 energy points for E LAB ∈ [600 , 2300] MeV γ [M. Gottschall et al., Phys. Rev. Lett. 112 no. 1, 012003 (2014)] - G : 19 energy points for E LAB ∈ [630 , 1950] MeV γ [A. Thiel et al., Phys. Rev. Lett. 109, 102001 (2012)] → Datasets overlap on 8 (!) energy-points E LAB ∈ [650 , 950] MeV! γ Y. Wunderlich Complete Experiments

  19. Description of the fitted datasets The following datasets were investigated for γ p → π 0 p : { σ 0 , Σ , T , P , E , G , H } . From investigations of the angular distributions of the data (and later confirmed by χ 2 / ndf in the multipole fit): ℓ max = 2 and ℓ max = 3 truncation approximations can already describe the data. Y. Wunderlich Complete Experiments

  20. Results of fully un-constrained analyses ℓ max = 2-fit ( N MC = 16000) ℓ max = 3-fit ( N MC = 24000) I. χ 2 -plot: I. χ 2 -plot: - There exists a global - There exists a global minimum. minimum. Y. Wunderlich Complete Experiments

  21. Results of fully un-constrained analyses ℓ max = 3-fit ( N MC = 24000) ℓ max = 2-fit ( N MC = 16000) I. χ 2 -plot: I. χ 2 -plot: - There exists a global - There exists a global minimum. minimum. - Local minima (ambiguities) - Global min. is well separated exists that have a very similar from other local minima. χ 2 to the global min. Y. Wunderlich Complete Experiments

  22. Results of fully un-constrained analyses ℓ max = 2-fit ( N MC = 16000) ℓ max = 3-fit ( N MC = 24000) I. blank I. blank II. Example multipole: II. Example multipole: : BnGa 2014-02 : ˆ σ ( W )-interval Y. Wunderlich Complete Experiments

  23. Results of fully un-constrained analyses ℓ max = 2-fit ( N MC = 16000) ℓ max = 3-fit ( N MC = 24000) I. blank I. blank II. Example multipole: � II. Example multipole: � χ 2 χ 2 � best / ndf + 0 . 5 � best / ndf + 0 . 05 : BnGa 2014-02 : ˆ σ ( W )-interval Y. Wunderlich Complete Experiments

  24. Results of fully un-constrained analyses ℓ max = 2-fit ( N MC = 16000) ℓ max = 3-fit ( N MC = 24000) I. blank I. blank II. Example multipole: � II. Example multipole: � χ 2 χ 2 � best / ndf + 1 . 0 � best / ndf + 0 . 3 : BnGa 2014-02 : ˆ σ ( W )-interval Y. Wunderlich Complete Experiments

  25. Results of fully un-constrained analyses ℓ max = 2-fit ( N MC = 16000) ℓ max = 3-fit ( N MC = 24000) I. blank I. blank II. blank II. blank III. Issues: III. Issues: - Over-fitting for lower E γ - Under-fitting for higher E γ - Accidential ambiguities: - Missing the � S , F � -, � P , F � - 4 2 × 3 − 1 max = 1 N AC � � = 2047. and � D , F � -interferences. 2 Y. Wunderlich Complete Experiments

  26. Results of fully un-constrained analyses ℓ max = 2-fit ( N MC = 16000) ℓ max = 3-fit ( N MC = 24000) I. blank I. blank II. blank II. blank III. Issues: III. Issues: - Over-fitting for lower E γ - Under-fitting for higher E γ - Accidential ambiguities: - Missing the � S , F � -, � P , F � - 4 2 × 3 − 1 max = 1 N AC � � = 2047. and � D , F � -interferences. 2 → 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

  27. χ 2 ndf vs. E γ for the fit including BnGa- F -waves Y. Wunderlich Complete Experiments

  28. χ 2 ndf vs. E γ for the fit including BnGa- F -waves Y. Wunderlich Complete Experiments

  29. The best solution for S -, P - and D -waves : BnGa 2014-02 : ˆ σ ( W )-interval Y. Wunderlich Complete Experiments

  30. The best solution for S -, P - and D -waves : BnGa 2014-02 : ˆ σ ( W )-interval Y. Wunderlich Complete Experiments

  31. � χ 2 � S -, P - and D -waves in the interval best + 1 . 0 : BnGa 2014-02 : ˆ σ ( W )-interval Y. Wunderlich Complete Experiments

  32. � χ 2 � S -, P - and D -waves in the interval best + 1 . 0 : BnGa 2014-02 : ˆ σ ( W )-interval Y. Wunderlich Complete Experiments

  33. Results for the S -, P - and D -waves 12 0.6 0.2 C � m Fm � C � m Fm � C � m Fm � 0.4 0.0 10 0.2 � 0.2 8 0.0 Im E 1 � ReE 0 � ReE 1 � � 0.4 � 0.2 6 � 0.6 � 0.4 � 0.8 700 750 800 850 900 700 750 800 850 900 700 750 800 850 900 E Γ � MeV � E Γ � MeV � E Γ � MeV � 8 5 C � m Fm � C � m Fm � C � m Fm � 2.0 7 4 6 1.5 3 5 1.0 2 Im M 1 � 4 ReM 1 � ReM 1 � 1 3 0.5 2 0 0.0 1 700 750 800 850 900 700 750 800 850 900 700 750 800 850 900 E Γ � MeV � E Γ � MeV � E Γ � MeV � C � m Fm � 2 Errors: bootstrapping 1 Im M 1 � 0 : BnGa 2014-02 � 1 : MAID-07 700 750 800 850 900 : SAID CM12 E Γ � MeV � Y. Wunderlich Complete Experiments

  34. Results for the S -, P - and D -waves 0.35 0.1 6 C � m Fm � C � m Fm � C � m Fm � 0.30 0.0 5 0.25 4 � 0.1 0.20 3 Im E 2 � ReE 2 � � 0.2 ReE 2 � 0.15 2 � 0.3 0.10 1 0 0.05 � 0.4 700 750 800 850 900 700 750 800 850 900 700 750 800 850 900 E Γ � MeV � E Γ � MeV � E Γ � MeV � 6 C � m Fm � 0.2 C � m Fm � C � m Fm � 0.2 4 0.1 0.0 0.0 2 � 0.2 � 0.1 Im M 2 � Im E 2 � ReM 2 � 0 � 0.4 � 0.2 � 0.3 � 2 � 0.6 700 750 800 850 900 700 750 800 850 900 700 750 800 850 900 E Γ � MeV � E Γ � MeV � E Γ � MeV � 0.5 3.5 C � m Fm � C � m Fm � 0.0 3.0 Errors: bootstrapping � 0.5 2.5 � 1.0 2.0 ReM 2 � Im M 2 � � 1.5 1.5 : BnGa 2014-02 � 2.0 1.0 � 2.5 : MAID-07 0.5 700 750 800 850 900 700 750 800 850 900 : SAID CM12 E Γ � MeV � E Γ � MeV � Y. Wunderlich Complete Experiments

  35. Summary & Outlook ∗ ) A monte-carlo sampling fit method was applied to { σ 0 , Σ , T , P , E , G , H } -data for γ p → π 0 p in the 2 nd resonance region. Y. Wunderlich Complete Experiments

  36. Summary & Outlook ∗ ) A monte-carlo sampling fit method was applied to { σ 0 , Σ , T , P , E , G , H } -data for γ p → π 0 p in the 2 nd resonance region. → ”LFits“ suggest an ℓ max = 2 (or 3)-truncation to describe the data. → ℓ max = 2 multipole fit: the best solution is ”unique“ but χ 2 too large (high-low partial wave interferences!) → ℓ max = 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. χ 2 / ndf and the behaviour of the solution are reasonable. Y. Wunderlich Complete Experiments

  37. Summary & Outlook ∗ ) A monte-carlo sampling fit method was applied to { σ 0 , Σ , T , P , E , G , H } -data for γ p → π 0 p in the 2 nd resonance region. → ”LFits“ suggest an ℓ max = 2 (or 3)-truncation to describe the data. → ℓ max = 2 multipole fit: the best solution is ”unique“ but χ 2 too large (high-low partial wave interferences!) → ℓ max = 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. χ 2 / 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 S varc): yields D 13 -pole-parameters � → Iteration of multipole-fitting with BnGa-code applied to SE-results: under construction ... Y. Wunderlich Complete Experiments

  38. Thank You!

  39. Additional Slides

  40. Details on the multipole fit procedure II σ ( W ). Example: ℓ ≤ ℓ max = 1, Ansatz: Use the total cross section ˆ � E C � � E C � phase constraint Im = 0 & Re > 0: 0+ 0+ � 2 + 6 Re � 2 + 6 Im � 2 + 2 Re � � 2 σ ( W ) ≈ 4 π q E C E C E C M C � � � � ˆ Re 0+ 1+ 1+ 1+ k � 2 + Re � 2 + Im � 2 � � M C � M C � M C +2 Im 1+ 1 − 1 − Y. Wunderlich Complete Experiments

  41. Details on the multipole fit procedure II σ ( W ). Example: ℓ ≤ ℓ max = 1, Ansatz: Use the total cross section ˆ � E C � � E C � phase constraint Im = 0 & Re > 0: 0+ 0+ � 2 + 6 Re � 2 + 6 Im � 2 + 2 Re � � 2 σ ( W ) ≈ 4 π q E C E C E C M C � � � � ˆ Re 0+ 1+ 1+ 1+ k � 2 + Re � 2 + Im � 2 � � M C � M C � M C +2 Im 1+ 1 − 1 − ∗ ) ˆ σ ( W ) constrains the intervals of the multipoles: � � � � � � � ˆ σ ( W ) ˆ σ ( W ) σ ( W ) ˆ � E C � k � M C � k k ∈ 0 , , . . . , Im ∈ − 4 π , Re 0+ 1 − q 4 π q q 4 π ∗ ) The total cross section, being quadratic form in the multipoles, also defines an ellipsoid in the multipole space. Y. Wunderlich Complete Experiments

  42. Cut selections for solution “data” χ 2 j − χ 2 Cut on solutions χ 2 best j with < ǫ ndf Mathematical ambiguity Unique best solution χ 2 χ 2 M ℓ M ℓ The χ 2 is defined by the fitted Legendre coefficients � α a Fit � k . L Y. Wunderlich Complete Experiments

  43. Cut selections for solution “data” χ 2 j − χ 2 Cut on solutions χ 2 best j with < ǫ ndf Mathematical ambiguity Unique best solution χ 2 χ 2 M ℓ M ℓ Start values have been distributed on the relevant part of the space M ℓ . Y. Wunderlich Complete Experiments

  44. Cut selections for solution “data” χ 2 j − χ 2 Cut on solutions χ 2 best j with < ǫ ndf Mathematical ambiguity Unique best solution χ 2 χ 2 M ℓ M ℓ Minimizations of χ 2 converge within several iterations. Y. Wunderlich Complete Experiments

  45. Cut selections for solution “data” χ 2 j − χ 2 Cut on solutions χ 2 best j with < ǫ ndf Mathematical ambiguity Unique best solution χ 2 χ 2 M ℓ M ℓ Minimizations of χ 2 converge within several iterations. Y. Wunderlich Complete Experiments

  46. Cut selections for solution “data” χ 2 j − χ 2 Cut on solutions χ 2 best j with < ǫ ndf Mathematical ambiguity Unique best solution χ 2 χ 2 M ℓ M ℓ Minimizations of χ 2 converge within several iterations. Y. Wunderlich Complete Experiments

  47. Cut selections for solution “data” χ 2 j − χ 2 Cut on solutions χ 2 best j with < ǫ ndf Mathematical ambiguity Unique best solution χ 2 χ 2 M ℓ M ℓ Minimizations of χ 2 converge within several iterations. Y. Wunderlich Complete Experiments

  48. Cut selections for solution “data” χ 2 j − χ 2 Cut on solutions χ 2 best j with < ǫ ndf Mathematical ambiguity Unique best solution χ 2 χ 2 M ℓ M ℓ Cut selection using ǫ = 1 Y. Wunderlich Complete Experiments

  49. Cut selections for solution “data” χ 2 j − χ 2 Cut on solutions χ 2 best j with < ǫ ndf Mathematical ambiguity Unique best solution χ 2 χ 2 M ℓ M ℓ Cut selection using ǫ = 1 / Cut selection using ǫ ∼ num . precision Y. Wunderlich Complete Experiments

  50. Bootstrapping ∗ ) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)] : Estimate an unknown � � X 1 , . . . , X n , ˆ distribution function of a random variable R F , by generating bootstrap random samples x b = ( x ∗ 1 , . . . , x ∗ n ) from the data � � x b , ˆ ( x 1 , . . . , x n ) and approximating the R -distribution-fct. by R b . F Y. Wunderlich Complete Experiments

  51. Bootstrapping ∗ ) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)] : Estimate an unknown � � X 1 , . . . , X n , ˆ distribution function of a random variable R F , by generating bootstrap random samples x b = ( x ∗ 1 , . . . , x ∗ n ) from the data � � x b , ˆ ( x 1 , . . . , x n ) and approximating the R -distribution-fct. by R b . F Y. Wunderlich Complete Experiments

  52. Bootstrapping ∗ ) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)] : Estimate an unknown � � X 1 , . . . , X n , ˆ distribution function of a random variable R F , by generating bootstrap random samples x b = ( x ∗ 1 , . . . , x ∗ n ) from the data � � x b , ˆ ( x 1 , . . . , x n ) and approximating the R -distribution-fct. by R b . F Y. Wunderlich Complete Experiments

  53. Bootstrapping ∗ ) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)] : Estimate an unknown � � X 1 , . . . , X n , ˆ distribution function of a random variable R F , by generating bootstrap random samples x b = ( x ∗ 1 , . . . , x ∗ n ) from the data � � x b , ˆ ( x 1 , . . . , x n ) and approximating the R -distribution-fct. by R b F . - Use gaussian distribution function centered at Ω α , with σ = ∆Ω α . → Resample data with replacement. → Ensemble of (1 + N Ens . ) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter. Y. Wunderlich Complete Experiments

  54. Bootstrapping ∗ ) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)] : Estimate an unknown � � X 1 , . . . , X n , ˆ distribution function of a random variable R F , by generating bootstrap random samples x b = ( x ∗ 1 , . . . , x ∗ n ) from the data � � x b , ˆ ( x 1 , . . . , x n ) and approximating the R -distribution-fct. by R b F . N Ens . = 1 - Use gaussian distribution function centered at Ω α , with σ = ∆Ω α . → Resample data with replacement. → Ensemble of (1 + N Ens . ) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter. Y. Wunderlich Complete Experiments

  55. Bootstrapping ∗ ) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)] : Estimate an unknown � � X 1 , . . . , X n , ˆ distribution function of a random variable R F , by generating bootstrap random samples x b = ( x ∗ 1 , . . . , x ∗ n ) from the data � � x b , ˆ ( x 1 , . . . , x n ) and approximating the R -distribution-fct. by R b F . N Ens . = 5 - Use gaussian distribution function centered at Ω α , with σ = ∆Ω α . → Resample data with replacement. → Ensemble of (1 + N Ens . ) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter. Y. Wunderlich Complete Experiments

  56. Bootstrapping ∗ ) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)] : Estimate an unknown � � X 1 , . . . , X n , ˆ distribution function of a random variable R F , by generating bootstrap random samples x b = ( x ∗ 1 , . . . , x ∗ n ) from the data � � x b , ˆ ( x 1 , . . . , x n ) and approximating the R -distribution-fct. by R b F . N Ens . = 10 - Use gaussian distribution function centered at Ω α , with σ = ∆Ω α . → Resample data with replacement. → Ensemble of (1 + N Ens . ) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter. Y. Wunderlich Complete Experiments

  57. Bootstrapping ∗ ) [B. Efron, The Annals Of Statistics 7 no. 1, 1 (1979)] : Estimate an unknown � � X 1 , . . . , X n , ˆ distribution function of a random variable R F , by generating bootstrap random samples x b = ( x ∗ 1 , . . . , x ∗ n ) from the data � � x b , ˆ ( x 1 , . . . , x n ) and approximating the R -distribution-fct. by R b F . N Ens . = 400 - Use gaussian distribution function centered at Ω α , with σ = ∆Ω α . → Resample data with replacement. → Ensemble of (1 + N Ens . ) bootstrap samples. Do TPWA for each. → Histogram results for each multipole-fit-parameter. Y. Wunderlich Complete Experiments

  58. Bootstrap histograms for E γ = 883 . 8 MeV C � m Fm � C � m Fm � C � m Fm � ReE 0 � ReE 1 � Im E 1 � 2.0 4 10 1.5 8 3 6 1.0 2 4 0.5 1 2 0.0 0 0 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 � 0.4 � 0.3 � 0.2 � 0.1 0.0 0.1 0.2 � 0.60 � 0.55 � 0.50 � 0.45 � 0.40 C � m Fm � C � m Fm � C � m Fm � ReM 1 � Im M 1 � ReM 1 � 4 4 1.5 3 3 1.0 2 2 0.5 1 1 0 0 0.0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 � 0.6 � 0.4 � 0.2 0.0 3.0 3.5 4.0 4.5 C � m Fm � Im M 1 � 2.5 2.0 1.5 1.0 0.5 0.0 � 0.8 � 0.6 � 0.4 � 0.2 0.0 0.2 Y. Wunderlich Complete Experiments

  59. Bootstrap histograms for E γ = 883 . 8 MeV C � m Fm � C � m Fm � C � m Fm � ReE 2 � Im E 2 � ReE 2 � 30 3.5 8 25 3.0 2.5 20 6 2.0 15 4 1.5 10 1.0 2 5 0.5 0 0 0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 � 0.04 � 0.020.00 0.02 0.04 0.06 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 C � m Fm � C � m Fm � C � m Fm � Im E 2 � ReM 2 � Im M 2 � 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 4.304.354.404.454.504.554.604.65 � 0.15 � 0.10 � 0.050.00 0.05 0.10 0.15 � 0.20 � 0.15 � 0.10 � 0.050.00 0.05 0.10 C � m Fm � C � m Fm � ReM 2 � Im M 2 � 5 4 4 3 3 2 2 1 1 0 0 1.7 1.8 1.9 2.0 2.1 2.2 2.3 � 1.1 � 1.0 � 0.9 � 0.8 � 0.7 � 0.6 Y. Wunderlich Complete Experiments

  60. Bootstrap histograms for E γ = 883 . 8 MeV C � m Fm � C � m Fm � C � m Fm � ReE 0 � ReE 1 � Im E 1 � 2.0 4 10 1.5 8 3 6 1.0 2 4 0.5 1 2 0.0 0 0 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 � 0.4 � 0.3 � 0.2 � 0.1 0.0 0.1 0.2 � 0.60 � 0.55 � 0.50 � 0.45 � 0.40 C � m Fm � C � m Fm � C � m Fm � ReM 1 � Im M 1 � ReM 1 � 4 4 1.5 3 3 1.0 2 2 0.5 1 1 0 0 0.0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 � 0.6 � 0.4 � 0.2 0.0 3.0 3.5 4.0 4.5 C � m Fm � Im M 1 � 2.5 2.0 1.5 1.0 0.5 0.0 � 0.8 � 0.6 � 0.4 � 0.2 0.0 0.2 Y. Wunderlich Complete Experiments

  61. Bootstrap histograms for E γ = 883 . 8 MeV C � m Fm � C � m Fm � C � m Fm � ReE 2 � Im E 2 � ReE 2 � 30 3.5 8 25 3.0 2.5 20 6 2.0 15 4 1.5 10 1.0 2 5 0.5 0 0 0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 � 0.04 � 0.020.00 0.02 0.04 0.06 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 C � m Fm � C � m Fm � C � m Fm � Im E 2 � ReM 2 � Im M 2 � 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 4.304.354.404.454.504.554.604.65 � 0.15 � 0.10 � 0.050.00 0.05 0.10 0.15 � 0.20 � 0.15 � 0.10 � 0.050.00 0.05 0.10 C � m Fm � C � m Fm � ReM 2 � Im M 2 � 5 4 4 3 3 2 2 1 1 0 0 1.7 1.8 1.9 2.0 2.1 2.2 2.3 � 1.1 � 1.0 � 0.9 � 0.8 � 0.7 � 0.6 Y. Wunderlich Complete Experiments

  62. Bootstrap histograms for E γ = 883 . 8 MeV C � m Fm � C � m Fm � C � m Fm � ReE 0 � ReE 1 � Im E 1 � 2.0 4 10 1.5 8 3 6 1.0 2 4 0.5 1 2 0.0 0 0 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 � 0.4 � 0.3 � 0.2 � 0.1 0.0 0.1 0.2 � 0.60 � 0.55 � 0.50 � 0.45 � 0.40 C � m Fm � C � m Fm � C � m Fm � ReM 1 � Im M 1 � ReM 1 � 4 4 1.5 3 3 1.0 2 2 0.5 1 1 0 0 0.0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 � 0.6 � 0.4 � 0.2 0.0 3.0 3.5 4.0 4.5 C � m Fm � Im M 1 � 2.5 2.0 1.5 1.0 0.5 0.0 � 0.8 � 0.6 � 0.4 � 0.2 0.0 0.2 Y. Wunderlich Complete Experiments

  63. Bootstrap histograms for E γ = 883 . 8 MeV C � m Fm � C � m Fm � C � m Fm � ReE 2 � Im E 2 � ReE 2 � 30 3.5 8 25 3.0 2.5 20 6 2.0 15 4 1.5 10 1.0 2 5 0.5 0 0 0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 � 0.04 � 0.020.00 0.02 0.04 0.06 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 C � m Fm � C � m Fm � C � m Fm � Im E 2 � ReM 2 � Im M 2 � 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 4.304.354.404.454.504.554.604.65 � 0.15 � 0.10 � 0.050.00 0.05 0.10 0.15 � 0.20 � 0.15 � 0.10 � 0.050.00 0.05 0.10 C � m Fm � C � m Fm � ReM 2 � Im M 2 � 5 4 4 3 3 2 2 1 1 0 0 1.7 1.8 1.9 2.0 2.1 2.2 2.3 � 1.1 � 1.0 � 0.9 � 0.8 � 0.7 � 0.6 Y. Wunderlich Complete Experiments

  64. Bootstrap histograms for E γ = 883 . 8 MeV C � m Fm � C � m Fm � C � m Fm � ReE 0 � ReE 1 � Im E 1 � 2.0 4 10 1.5 8 3 6 1.0 2 4 0.5 1 2 0.0 0 0 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 � 0.4 � 0.3 � 0.2 � 0.1 0.0 0.1 0.2 � 0.60 � 0.55 � 0.50 � 0.45 � 0.40 C � m Fm � C � m Fm � C � m Fm � ReM 1 � Im M 1 � ReM 1 � 4 4 1.5 3 3 1.0 2 2 0.5 1 1 0 0 0.0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 � 0.6 � 0.4 � 0.2 0.0 3.0 3.5 4.0 4.5 C � m Fm � Im M 1 � 2.5 2.0 1.5 1.0 0.5 0.0 � 0.8 � 0.6 � 0.4 � 0.2 0.0 0.2 Y. Wunderlich Complete Experiments

  65. Bootstrap histograms for E γ = 883 . 8 MeV C � m Fm � C � m Fm � C � m Fm � ReE 2 � Im E 2 � ReE 2 � 30 3.5 8 25 3.0 2.5 20 6 2.0 15 4 1.5 10 1.0 2 5 0.5 0 0 0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 � 0.04 � 0.020.00 0.02 0.04 0.06 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 C � m Fm � C � m Fm � C � m Fm � Im E 2 � ReM 2 � Im M 2 � 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 4.304.354.404.454.504.554.604.65 � 0.15 � 0.10 � 0.050.00 0.05 0.10 0.15 � 0.20 � 0.15 � 0.10 � 0.050.00 0.05 0.10 C � m Fm � C � m Fm � ReM 2 � Im M 2 � 5 4 4 3 3 2 2 1 1 0 0 1.7 1.8 1.9 2.0 2.1 2.2 2.3 � 1.1 � 1.0 � 0.9 � 0.8 � 0.7 � 0.6 Y. Wunderlich Complete Experiments

  66. Polarization Observables - more detail Problem: 4 complex amplitudes F i ( W , θ ) ≡ 8 real numbers � d σ � ⇒ 1 observable 0 insufficient to determine the amplitudes! d Ω Y. Wunderlich Complete Experiments

  67. Polarization Observables - more detail Problem: 4 complex amplitudes F i ( W , θ ) ≡ 8 real numbers � d σ � ⇒ 1 observable 0 insufficient to determine the amplitudes! d Ω Solution: Utilize the polarization degrees of freedom of the reaction Example: Beam- [A. Thiel, PhD (2012)] Polarization E γ = 1000 MeV � � d σ d Ω ⇒ φ N ∝ ← reaction plane φ [ deg ] Y. Wunderlich Complete Experiments

  68. Polarization Observables - more detail Problem: 4 complex amplitudes F i ( W , θ ) ≡ 8 real numbers � d σ � ⇒ 1 observable 0 insufficient to determine the amplitudes! d Ω Solution: Utilize the polarization degrees of freedom of the reaction Example: Beam- [A. Thiel, PhD (2012)] Polarization E γ = 1000 MeV � � d σ d Ω ⇒ φ N ∝ ← reaction plane φ [ deg ] The observable Σ appears as amplitude of the φ -modulation � d σ � d σ � � ( θ, φ ) = 0 (1 − ǫ L Σ cos(2 φ )). d Ω d Ω Σ is an asymmetry between different polarization states: � ( ⊥ , 0 , 0) − �� d σ � d σ � ( � , 0 , 0) � 1 Σ = . 2 ( d σ d Ω ) 0 d Ω d Ω Y. Wunderlich Complete Experiments

  69. Complete Experiment: truncated partial wave analysis ∗ ) Question: How many and which observables are needed if multipoles { E ℓ ± , M ℓ ± } are the goal in a PWA truncated at some ℓ max ?. Y. Wunderlich Complete Experiments

  70. 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 b 4 ( θ ) = b 3 ( − θ ), b 2 ( θ ) = b 1 ( − θ ) and do a linear factor decomposition of b 2 and b 4 : i θ � � exp 2 b 2 ( θ ) = −C a 2 ℓ max (1 + t 2 ) ℓ max [( t − β 1 ) ( t − β 2 ) . . . ( t − β 2 ℓ max )] i θ � � exp 2 b 4 ( θ ) = C a 2 ℓ max (1 + t 2 ) ℓ max [( t − α 1 ) ( t − α 2 ) . . . ( t − α 2 ℓ max )] → A set of Omelaenko-roots { α k , β k } is fully equivalent to a multipole-solution { E ℓ ± , M ℓ ± } . Y. Wunderlich Complete Experiments

  71. Complete Experiment: truncated partial wave analysis ∗ ) Important hint: [A.S. Omelaenko, Sov. J. Nucl. Phys. 34, 406 (1981).] exp ( i θ 2 ) b 2 ( θ ) = −C a 2 ℓ max (1+ t 2 ) ℓ max [( t − β 1 ) ( t − β 2 ) . . . ( t − β 2 ℓ max )] exp ( i θ 2 ) b 4 ( θ ) = C a 2 ℓ max (1+ t 2 ) ℓ max [( t − α 1 ) ( t − α 2 ) . . . ( t − α 2 ℓ max )] ∗ ) Complex conjugation of all { α k , β k } , or some subset of them, leaves | b i | 2 invariant and therefore also the group S � ± | b 1 | 2 ± | b 2 | 2 ± | b 3 | 2 + | b 4 | 2 � σ 0 , ˇ Σ , ˇ T , ˇ ≡ ˇ Ω α S = 1 � � P 2 Y. Wunderlich Complete Experiments

  72. Complete Experiment: truncated partial wave analysis ∗ ) Important hint: [A.S. Omelaenko, Sov. J. Nucl. Phys. 34, 406 (1981).] exp ( i θ 2 ) b 2 ( θ ) = −C a 2 ℓ max (1+ t 2 ) ℓ max [( t − β 1 ) ( t − β 2 ) . . . ( t − β 2 ℓ max )] exp ( i θ 2 ) b 4 ( θ ) = C a 2 ℓ max (1+ t 2 ) ℓ max [( t − α 1 ) ( t − α 2 ) . . . ( t − α 2 ℓ max )] ∗ ) Complex conjugation of all { α k , β k } , or some subset of them, leaves | b i | 2 invariant and therefore also the group S � ± | b 1 | 2 ± | b 2 | 2 ± | b 3 | 2 + | b 4 | 2 � σ 0 , ˇ Σ , ˇ T , ˇ ≡ ˇ Ω α S = 1 � � P 2 � � ∗ ) Omelaenko’s constraint β k ′ has to be fulfilled. α k = k ′ k Y. Wunderlich Complete Experiments

  73. Complete Experiment: truncated partial wave analysis ∗ ) Important hint: [A.S. Omelaenko, Sov. J. Nucl. Phys. 34, 406 (1981).] exp ( i θ 2 ) b 2 ( θ ) = −C a 2 ℓ max (1+ t 2 ) ℓ max [( t − β 1 ) ( t − β 2 ) . . . ( t − β 2 ℓ max )] exp ( i θ 2 ) b 4 ( θ ) = C a 2 ℓ max (1+ t 2 ) ℓ max [( t − α 1 ) ( t − α 2 ) . . . ( t − α 2 ℓ max )] σ 0 , ˇ Σ , ˇ T , ˇ � � ∗ ) Complex conjugation of { α k , β k } leaves P invariant. ∗ ) Omelaenko’s constraint � k α k = � k ′ β k ′ . → 2 kinds of symmetries/ambiguities: accidential ambiguities double ambiguity - α k → α ∗ k , β k → β ∗ - Conjugation of subset of { α k , β k } k for all k α k , ˜ � � → ˜ β k - One has always � k α k = � k ′ β k ′ k ′ ˜ → � k α ∗ k = � k ′ β ∗ - Very (very) likely � α k ≃ � k ˜ β k ′ k ′ → Mathematically exact symmetry → Manifest as approximate symmetry - Is resolved by F , G , as well as any - Is resolved by in principle any BR and T R observable. observable. Y. Wunderlich Complete Experiments

  74. Which ℓ max to choose? → “LFit-method” ∗ ) Utilize the parametrization of the angular distributions of polarization Ω α as expansions into P m observables ˇ ℓ (cos θ ) for fixed energy: 2 ℓ max + β α + γ α Ω α ( W , θ ) = k ( W ) P β α ˇ � ( a L ) α (cos θ ) . k k = β α Y. Wunderlich Complete Experiments

  75. Which ℓ max to choose? → “LFit-method” ∗ ) Utilize the parametrization of the angular distributions of polarization Ω α as expansions into P m observables ˇ ℓ (cos θ ) for fixed energy: 2 ℓ max + β α + γ α Ω α ( W , θ ) = k ( W ) P β α ˇ � ( a L ) α (cos θ ) . k k = β α → Fit angular distributions with some low initial ℓ max ( ℓ max = 0 most commonly) and see if χ 2 / ndf is satifactory. If not: χ 2 / ndf � � → Raise truncation order by 1 and do new fit until ≈ 1. → Hint for dominant partial wave by the order ℓ max at which this procedure terminates. Y. Wunderlich Complete Experiments

  76. Which ℓ max to choose? → “LFit-method” ∗ ) Utilize the parametrization of the angular distributions of polarization Ω α as expansions into P m observables ˇ ℓ (cos θ ) for fixed energy: 2 ℓ max + β α + γ α Ω α ( W , θ ) = k ( W ) P β α ˇ � ( a L ) α (cos θ ) . k k = β α → Fit angular distributions with some low initial ℓ max ( ℓ max = 0 most commonly) and see if χ 2 / ndf is satifactory. If not: χ 2 / ndf � � → Raise truncation order by 1 and do new fit until ≈ 1. → Hint for dominant partial wave by the order ℓ max 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

  77. LFits to { σ 0 , Σ , T , P , E , G , H } To be published in [Y. W., F. Afzal, A. Thiel and R. Beck, (2016)] ˇ σ 0 Σ E [MeV] E [MeV] γ γ 500 1000 1500 600 800 1000 1200 1400 40 /ndf /ndf L =1 L =1 2 max 2 χ χ max L =2 max L =2 y y max 30 L =3 40 r r max a a L =3 L =4 max n n max i i L =4 L =5 m m max max i i 20 l l e e r r P P 20 10 0 0 1200 1400 1600 1800 1400 1600 1800 ˇ W [MeV] ˇ W [MeV] T P E [MeV] E [MeV] γ γ 1000 1500 700 800 900 6 /ndf /ndf L =1 L =1 2 2 χ max χ max L =2 L =2 y y max max 20 r r L =3 L =3 a a max 4 max n n L =4 L =4 i i m m max max i i l l e e r r P P 10 2 0 0 1600 1800 2000 1500 1550 1600 W [MeV] W [MeV] Y. Wunderlich Complete Experiments

  78. LFits to { σ 0 , Σ , T , P , E , G , H } To be published in [Y. W., F. Afzal, A. Thiel and R. Beck, (2016)] ˇ ˇ E G E [MeV] E [MeV] γ γ 1000 1500 2000 800 1000 1200 20 /ndf /ndf 20 L =1 L =1 2 2 χ max χ max L =2 L =2 y max y max 15 r r a L =3 a L =3 15 max max n n i L =4 i L =4 m max m max i i 10 l l e e 10 r r P P 5 5 0 0 1600 1800 2000 2200 1500 1600 1700 1800 ˇ W [MeV] W [MeV] H E [MeV] γ 700 800 900 2.5 /ndf L =1 Overall, ℓ max = 2 should be OK 2 χ max 2 L =2 y max r L =3 a max in all energy bins n L =4 1.5 i m max i l e E LAB ∈ [650 , 950] MeV r P γ 1 except maybe the last 2 bins. 0.5 0 1500 1550 1600 W [MeV] Y. Wunderlich Complete Experiments

  79. χ 2 best vs. E γ for the ℓ max = 2-fit Y. Wunderlich Complete Experiments

  80. χ 2 best vs. E γ for the ℓ max = 2-fit Y. Wunderlich Complete Experiments

  81. The best solution for S -, P - and D -waves Y. Wunderlich Complete Experiments

  82. The best solution for S -, P - and D -waves Y. Wunderlich Complete Experiments

  83. � χ 2 � S -, P - and D -waves in the interval best + 0 . 5 Y. Wunderlich Complete Experiments

  84. � χ 2 � S -, P - and D -waves in the interval best + 0 . 5 Y. Wunderlich Complete Experiments

  85. � χ 2 � S -, P - and D -waves in the interval best + 1 . 0 Y. Wunderlich Complete Experiments

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