1. Poles of PWD ata and PWA mplitudes in Zagreb model A. Švarc, S. Ceci, B. Zauner Rudjer Bošković Institute, Zagreb, Croatia M. Hadžimehmedović, H. Osmanović, J. Stahov Univerzity of Tuzla, Tuzla Bosnia and Herzegovina M. Hadžimehmedović, H. Osmanović, J. Stahov Univerzity of Tuzla, Tuzla Bosnia and Herzegovina

2. How do I see what is our main aim? Experiment Quarks ? Matching point structures bound states resonances

3. Höhler – Landolt Bernstein 1984. Burkert – Lee 2004. Ceci, Svarc, Zauner 2005. Svarc 2004 Höhler – Landolt Bernstein 1984.

4. explicit analytic form introduced Phenomenological T-matrix o Phenomenological T-matrix CMU-LBLCMU-LBL ZagrebZagreb Argonne-PittsburghArgonne-Pittsburgh o effective Lagrangian EBAC EBAC JuelichJuelich Dubna-Mainz-Taipei (DMT)Dubna-Mainz-Taipei (DMT) GiessenGiessen o Chew-Mandelstam K-matrix GWU/VPIGWU/VPI PWD PWA Difference between PWD and PWA

5. How do I see what is our main aim? Experiment Quarks structures bound states resonances Breit-Wigner parameters Pole parameters Phase shifts

6. What is “better”: Breit-Wigner parameters Breit-Wigner parameters or or Pole parameters Pole parameters The advantages and drawbacks

7. Breit – Wigner parameters: Advantages: defined on the real axes defined on the real axes simple to calculate simple to calculate Harry Lee BRAG 2001 Drawbacks: dependence on the choice of field variablesdependence on the choice of field variables model dependent (background definition) model dependent (background definition)

8. A simple illustration of BW model dependence:

11. Pole parameters Advantages invariant with respect to the choice of field variables invariant with respect to the choice of field variables model independent model independent less model dependentless model dependent Drawbacks hard to get because they lie in the complex energy plane hard to get because they lie in the complex energy plane Why?

12. The dependence upon background parameterization is a well know fact, but nothing has been done in PDG yet. PDG makes an average of all BW values, regardless of the way background has been introduced. This introduces an additional systematic error. Extraction of Breit-Wigner parameters

13. Suggestion by Lothar Tiator: We all know that BW positions and parameters are not well defined but many people believe that within some uncertainty they can be given, and are very useful. Therefore I also tend to stick with them and would propose to keep them in PDG in the future. But only if we can give some proper definition and methods for extraction. Suggestion by Harry Lee:

14. Each PWA has a specific assumption on the analytic form. Idea: Let us use CMB formalism to analyze available PWD and PWA using one and the same analytic form. We propose: 1.To use new PWD or PWA in addition to the existing ones and look for the shift of poles (shift of present ones, appearance of the new ones) 2. To use ONE analytic form (Zagreb CMB) for ALL EXISTING PWA and PWD, and: a.extract poles from a particular PWD or PWA using Zagreb CMB and compare the outcome with the original result b.compare agreement of poles of ALL EXISTING PWA and PWD in order to avoid systematic error because of differences in analytic forms. Extraction of pole parameters

15. Warning: When we analyze particular PWA, we do not say that we shall exactly reproduce pole positions given by that particular model. Results may differ, and the difference will show the significance of analytic form chosen to represent on-energy shell data. So, we are not checking if the pole positions in a certain model are correctly extracted, but rather seting up the way how to quantify the comparison of different curves.

16. 1. … to use new PWD or PWA.... Initial attempts...

17. In details repeated at NSTAR2005 - Tallahassee Technical problems on Zagreb side......

18. 2.Using CMB to analyze a world collection of PWD and PWA and eliminate model assumptions on analytic form Formulated at BRAG2007 Formulated at BRAG2007 Research in progress Research in progress REMARK Technical problems in Zagreb code are now eliminated. Code is running under LINUX. It is transferable from machine to machine, and is an open source code. Adjustments and improvements can be done. (I can demonstrated how the code works during workshop)

19. All coupled channel models are based on solving Dyson-Schwinger integral type equations, and they all have the same general structure: full = bare + bare * interaction* full full = bare + bare * interaction* full CMB coupled-channel model

20. Carnagie-Melon-Berkely (CMB) model is an isobar model where Instead of solving Lipmann-Schwinger equation of the type: with microscopic description of interaction term we solve the equivalent Dyson-Schwinger equation for the Green function with representing the whole interaction term effectively.

21. We represent the full T-matrix in the form where the channel-resonance interaction is not calculated but effectively parameterized. Model is manifestly unitary and analytic. channel-resonance mixing matrix bare particle propagator channel propagator

22. Model assumptions: isobar model (poles are introduced as intermediate resonant states called intermediate particles)isobar model (poles are introduced as intermediate resonant states called intermediate particles) background parameterization - meromorphic function background parameterization - meromorphic function the form of imaginary part of the channel propagator introduces proper channel cutsthe form of imaginary part of the channel propagator introduces proper channel cuts where q a (s) is the meson-nucleon cms momentum: Imaginary part of the channel propagator is defined as:

23. q a (s) is the meson-nucleon cms momentum: The analyticity is manifestly imposed by calculating the channel propagator real part through the dispersion relation: The unitarity has been proven by Cutkosky.

24. The full solution is given as: =  ij T

25. we define the number of background poles we define the number of background poles we define the number of resonance poles we define the number of resonance poles we fit we fit s i......... resonance mass s i......... resonance mass  ic......... channel resonance mixing parameters  ic......... channel resonance mixing parameters Final result: energy dependent partial wave T-matrices on the real axes Step 1: Fitting procedure Step 2: Extracting resonance parameters – go into the complex energy plane (singularity structure of the obtained solution)

27. To find the position of poles of the matrix T(s) in the complex energy plane we have to find all zeroes of the denominator. So, we solve the following equation:

28. How do we solve it? when obtained from the fit, det G -1 (s) is a complex function of a real argument s when obtained from the fit, det G -1 (s) is a complex function of a real argument s we have to we have to 1. analytically continue this function into the complex energy plane (observe that only channel propagator  (s) has to be analytically continued 2. find a complex zero s 0 of that function in the complex energy plane – we do it numerically

29. 1.Analytical continuation of the channel propagator  (s) a.Numerical integration (In old paper) b.Nowadays - Pietarinen expansion We have constructed a function: Observe that this is a complex function of a complex argument function of a complex argument for physical argument x! for physical argument x! for x > x 0 x 0 – x is negative, and Z I (x) is complex for x > x 0 x 0 – x is negative, and Z I (x) is complex

30. How does it look in practice?

32. 2.Finding a complex zero We did it numerically: instead of calculating | det G -1 | we have calculated | det G | instead of calculating | det G -1 | we have calculated | det G | we made a 3D plot we made a 3D plot | det G | = f (Re s, Im s) | det G | = f (Re s, Im s) and numerically looked for the point of infinity of this function.

34. Mathematical operationalization

35. Mass → Width → Partial width →

36. Stability of the procedure Stability of the procedure has been tested with respect with different model assumptions of Zagreb CMB: 1.Defining the input 2.Form of the channel propagator (meson-resonance vertex function) Inner partInner part Asymptotic partAsymptotic part Cut off parametersCut off parameters 3.Type of the background 4.Number of channels 5.Mass of the effective channel (To be given at the end of the talk if time permits....)

37. Results a.Use Zagreb CMB fits to analyze a particular PWD or PWA b.Compare all PWA and PWD in Zagreb CMB in order to avoid systematic differences in analytic continuation

38. Importance of inelastic channels Elastic channels only are insufficient to constrain all T-matrix poles, especially those which dominantly couple to inelastic channels.

39. In reality - P 11 (1710) example

40. We use: 1.CMB model for 3 channels:  N,  N, and dummy channel     N elastic T matrices, PDG: SES Ar06  N   N T matrices, PDG: Batinic 95 We fit: 1.πN elastic only  N   N only 3.both channels

41. Results for extracted pole positions:

45. Use Zagreb CMB fits to analyze a particular PWD or PWA I. Dubna – Mainz – Taipei (DMT) S 11 : DMT model fits GWU/VPI single energy solutions, and obtains:

46. They also fit πN→ηN S 11 But we shall return to importance of elastic channels later.

47. Dilemma: How many dressed poles does one find in DMT functions? Facts of life: DMT model has 4 bare poles in S 11 DMT model has 4 bare poles in S 11 bare poles gets dressed, travel from the real axes into the complex energy plane, and there is no a priori way to say where they endbare poles gets dressed, travel from the real axes into the complex energy plane, and there is no a priori way to say where they end one needs either analytic continuation of DMT functions or some other pole search methodone needs either analytic continuation of DMT functions or some other pole search method up to now DMT uses speed plot techniqueup to now DMT uses speed plot technique

48. But find only 3 poles

49. We analyze their function with Zagreb CMB, and make a fit with 3 bare poles and 4 bare poles.

50. 3R 4R

51. This is a good place to illustrate the difference in analytic structure between CMB and other models. At the same time this is a good place to illustrate the model dependence of bare parameters. Both, Zagreb and DMT have the same structure full = bare + bare * interaction* full full = bare + bare * interaction* full and this brings us to the self energy, and the separation to the bare and dressed quantities. I will illustrate that the way of separation bare ↔ dressed quantities bare ↔ dressed quantities is different in DMT and Zagreb.

52. I take DMT amplitudes, fit them with Zagreb CMB, but fix bare Zagreb masses to DMT values: M 1 0 = 1559 MeV M 2 0 = 1727 MeV M 3 0 = 1803 MeV M 4 0 = 2090 MeV

54. M 1 0 = 1559 MeV M 2 0 = 1727 MeV M 3 0 = 1803 MeV M 4 0 = 2090 MeV M 1 0 = 1506 MeV M 2 0 = 1650 MeV M 3 0 = 1853 MeV M 4 0 = 2118 MeV S 1 = 1600 + i 105 S 2 = 1680 + i 750 S 3 = 1790 + i 105 S 4 = 2140 + i 350 S 1 = 1501 + i 118 S 2 = 1640 + i 124 S 3 = 1921 + i 176 S 4 = 2129 + i 224 Best fit Χ 2 red = 0.63 Χ 2 red = 0.07

55. Conclusion: We support 4 bare pole solution, but find the 4 th dressed pole too. We see three poles under 2 GeV and one above. The latest DMT analysis goes beyond speed plot, uses analytic continuation, and says something different: analytic continuation sees the pole where speed plot does not!

57. II. EBAC Original motivation: strong change in πN elastic PW when the ηN data are included Compare Zagreb CMB fits with particular PWD or PWA Diaz07 (πN elastic) PWA Durand08 (πN elastic + πN → ηN) PWA

58. Normalization of ηN PW? Guided by: In publication we find

59. We have analysed Diaz07 (πN elastic) PWA Diaz07 (πN elastic) PWA Durand08 (πN elastic + πN → ηN) PWA Durand08 (πN elastic + πN → ηN) PWA (as in publication) (as in publication) Durand08 (πN elastic + πN → ηN) PWA (renormalized as suggested by Dűring)Durand08 (πN elastic + πN → ηN) PWA (renormalized as suggested by Dűring)

60. Only πN elastic Diaz07 fitted

61. Only πN elastic Durand08 fitted

65. Renormalizatio suggested by Michael Dűring (December 2009) Phase space factor He replaced the PS factor for πN elastic channel with the PS factor for πN → ηN channel

66. So, he calculated the corerction factor N2/N1 and obtained

71. III. Juelich Compare Zagreb CMB fits with particular PWD or PWA πN elastic – 3R

72. πN elastic + πN→ηN

73. Compare all PWA and PWD in Zagreb CMB in order to avoid systematic differences in analytic continuation

74. πN→πN

75. KH80

77. πN→πN + πN→ηN

78. πN elastic

79. πN elastic + πN → ηN

80. PiN-PiN-3R

81. PiN-PiN-4R

82. PiN-EtaN-3R

83. PiN-EtaN-4R

84. PiN-PiN-3R

85. PiN-EtaN-3R

86. PiN-PiN-4R

87. PiN-EtaN-4R

88. Stability of the procedure Stability of the procedure has been tested with respect with different model assumptions of Zagreb CMB: 1.Defining the input 2.Form of the channel propagator (meson-resonance vertex function) Inner partInner part Asymptotic partAsymptotic part Cut off parametersCut off parameters 3.Type of the background 4.Number of channels 5.Mass of the effective channel

89. Asymptotic partAsymptotic part

90. Inner part (unchanged threshold behavior)Inner part (unchanged threshold behavior)

92. Inner part (threshold behavior changed)Inner part (threshold behavior changed)