A. Švarc Rudjer Bošković Institute, Zagreb, Croatia INT-09-3

1. Inherent model dependence of Breit-Wigner parameters (poles as a true signal of resonance properties) A. Švarc Rudjer Bošković Institute, Zagreb, Croatia INT-09-3 The Jefferson Laboratory Upgrade to 12 GeV (Thursday, November 12, 2009)

2. Batinic 95

3. Publications

4. Breit-Wigner parameters and poles are discussed within the scope of PWA - hadron spectroscopy What is hadron spectroscopy ? What is the aim of hadron spectroscopy?

5. What we actually do and who is doing what? BEFORE NOWADAYS Höhler – LandoltBernstein 1984.

6. And how are the results currentlypresented?

7. Pole positions Breit-Wigner parameters PDG2008

8. We have to ask ourselves two questions: What do we know about Breit-Wigner parameters? What do we know about poles?

9. What do we know about Breit-Wigner parameters? Breit-Wigner parameters are MODEL DEPENDENT! We know for decades: continuum ambiguities But UNFORTUNATELY this fact is not unanimously accepted!

10. A simple iIllustration of BW model dependence:

14. And what is the way out? Use another, more model independent quantity: as recommended in PDG: POLES

15. Can we do any recommendations here? • Before this morning I wanted to suggest: • Establish a consensus if this is really true • Write a note in the “white paper” • Ask PDG (C. Wohl) to restore the introductory part about Breit-Wigner model dependence • Suggest to PDG to change the order of appearance, to put the poles first • Write a paper in which one should show if there is any difference if pole positions are used instead of BW parameters, and if not where to expect the effect But now I believe that we should

16. What do we know about poles? Mandelstam hypothesis There is no fundamental differencebetweena bound state and a resonance, other than the matterof stability, it is to be expected that when simple polesof the coupled channel amplitude occur onunphysicalsheets in the complex energy plane, they ARE TO BEassociated with resonant states.

17. Let us see what our problem is What do we measure? Quantities on the physical axis What are we looking for? Poles in the complex energy plane Problem: How do we parameterize simple poles in the complex energy plane when we can only see data lying on the physical axis?

18. Our conclusion: Breit-Wigner parameters : model dependent their interpretation is not clear recommendation: do not extract without defining the procedure b) Poles are recommended

19. However in QCD ....

20. Questions appear: How many type of poles do we have? Are some of them “more fundamental” then the others? Are all of them the “image” of some internal structure ? • As physical observable can be expressed in terms of either T or K matrices, from ancient times we know that we have: • T – matrix poles • K – matrix poles • and from recently we know that we have • Bare poles

21. There are many ways of looking for the poles, but I would like to show one of them we are able to apply

22. Methods Breit-Wigner parameters • constant background • energy dependent background • Flatte Speed-Plot Regularizationmethod Time-delay N/D method Analytic continuation

23. The procedure to analyze the T matrix poles we present now was first proposed at NSTAR2005 in Tallahassee and elaborated at the BRAG2007 meeting in Bonn: As extracting poles is related to the exact form of the analytic function which is used to describe PW amplitudes, we propose to use only one method to extract pole positions from published partial wave analyses understanding them as nothing else but good, energy dependent representations of all analyzed experimental data – as partial wave data (PWD). We have chosen the3-channelT-matrix Carnegie-Melon-Berkeley (CMB) model for which we have developed our own set of codes and fitted “ALL AVAILABLE” PWD or PWA we could find “on the market”. What is the benefit? Alll errors due to different analytic continuations of different models are avoided, and the only remaining error is the precision of CMB method itself. Importanceofinelasticchannels IS CLEARLY VISIBLE

24. CMB coupled-channel model • 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

25. Carnagie-Melon-Berkely (CMB) model is anisobar 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.

26. We represent the full T-matrix in the form where the channel-resonance interaction is not calculated but effectively parameterized: bare particle propagator channel-resonance mixing matrix channel propagator

27. Assumption: The imaginary part of the channel propagator is defined as: And we require its analyticity through the dispersion relation: where qa(s) is the meson-nucleon cms momentum:

28. we obtain the full propagator G by solving Dyson-Schwinger equation where we obtain the final expression

29. The analysis is done for the S11 partial wave. Analysis of other partial wave is in progress. Input data π-N elastic, Imaginarypart

31. π-N elastic, Real part

33. πN elastic channel: • Gross features of ALL partial wave amplitudes are the same • “two peak structure” is always present • Minor differences in imaginary part: • Peak position is sometimes slightly shifted (second peak for Giessen) • Size of the peaks is different • Second peak is much narrower for Giessen • First peak is somewhat lower for Giessen • Second peak is much lower for EBAC • The structure of higher energy part is different (smooth for FA08, much more structure for others)

34. πN → ηN, Imaginarypart

36. πN → ηN, Real part

38. πN →ηN channel: • Gross features of ALL partial wave amplitudes are similar • “two peak structure” is visible • Notable differences in imaginary part: • Peak strength is much higher in EBAC • Second peak starts very early for Giessen, somewhat later for EBAC and much later for Zagreb • Strength of the higher energy peak is shifted to the real part for Zagreb

39. Present situation with poles. Poles in PDG2008:

45. Our solutions: • We have two sets of solutions: • We have fitted πN elastic channel only • We have fitted πN elastic + πN→ηN

46. πN elastic, imaginarypart Typical results: