1. Partial Wave and N* Analysis with Analyticity Alfred Svarc Rudjer Boskovic Institute, Zagreb, Croatia PWA8/ATHOS3 20151

2. 2 This is an old question….. Possible answers: backwards looping of Argand diagrambackwards looping of Argand diagram existence of time delayexistence of time delay scattering phase going through π/2scattering phase going through π/2 structure in speed plotstructure in speed plot pole in the partial wave scattering matrix ……………….pole in the partial wave scattering matrix ……………….

3. PWA8/ATHOS3 20153 Current (PDG) definition

4. PWA8/ATHOS3 20154

5. 5 SINGLE ENERGY (SE) (SE) ENERGY DEPENDENT (ED) (ED) SINGLE CHANNEL (SC) (SC) COUPLED CHANNEL (CC) (CC) The oldest Breit Wigner functions with energy independent and energy dependent mass and width ---- Elaborate reaction models EXPLICITLY ANALYTIC UNITARY IN ELASTIC REGIME Contemporary Elaborate reaction models EXPLICITLY ANALYTIC AND UNITARY Amplitude and PW reconstruction In principle NO UNITARITY AND ANALYTICITY Problem: AMBIGUITIES Solution: imposing A and U constraints Very little has been done What kind of PWA do we have?

6. PWA8/ATHOS3 20156 Analyticity Scattering matrix is an analytic function in Mandelstam s, t and u variables.

7. PWA8/ATHOS3 20157 Analyticity in ED PWA Usually implemented through analyticity of S-matrix functions.

8. PWA8/ATHOS3 20158 Analyticity in SE PWA Problem: How to implement analyticity because SE PWA is amplitude reconstruction in a discrete set of energy points? analytic penalty functionanalytic penalty function What is the choice of analytic penalty function? theoretical modelstheoretical models Pietarinen expansionPietarinen expansion

9. PWA8/ATHOS3 20159 Analyticity in s (W) (fixed W; usually exploited) Tiator, MAID collaboration meeting, Mainz 2015 PWA fixed W

10. PWA8/ATHOS3 201510 Analyticity in t = f(W,θ) (fixed t; very rarely mentioned!) Tiator, MAID collaboration meeting, Mainz 2015

11. PWA8/ATHOS3 201511 Analyticity in t = f(W,θ) Tiator, MAID collaboration meeting, Mainz 2015 PWA fixed t

12. PWA8/ATHOS3 201512 We still didn’t say how we introduce analyticity constraints!

13. PWA8/ATHOS3 201513 SE analysis is amplitude reconstruction in a discrete set of data points. To impose analyticity we use „penalty function” technique during minimization, so we have to define SOME analytic function to penalize to! Choice of analytic function: STANDARD: solution of a theoretical model Instead we use PIETARINEN EXPANSION!

14. PWA8/ATHOS3 201514 Pietarinen expansion SE SC PWASE SC PWA Extraction of poles from SE and ED partial wavesExtraction of poles from SE and ED partial waves New method (2013)

15. PWA8/ATHOS3 201515 If you create a model, the advantage is that your solution is absolutely global, valid in the full complex energy plane (all Rieman sheets). The drawback is that the solution is complicated, pole positions are usually energy dependent otherwise you cannot ensure simple physical requirements like absence of the poles on the first, physical Riemann sheet, Schwartz reflection principle, etc. It is complicated and demanding to solve it. THEORETICAL MODELS WE PROPOSE Construct an analytic function NOT in the full complex energy plane, but CLOSE to the real axes in the area of dominant nucleon resonances, which is fitting the data. Idea: GLOBALITY FOR SIMPLICITY TRADING ADVANTAGES

16. PWA8/ATHOS3 201516 Instead of „guessing” the exact form of used analytic function via theoretical models we EXPAND IT IN FASTLY CONVERGENT POWER SERIES OF PIETARINEN („Z”) FUNCTIONS WITH WELL KNOWN BRANCH-POINTS! Original idea: 1.S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961) 2.I. Ciulli, S. Ciulli, and J. Fisher, Nuovo Cimento 23, 1129 (1962). Convergence proven in: 1.S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961) 2.Detailed proof in I. Caprini and J. Fischer: "Expansion functions in perturbative QCD and the determination of α s ", Phys.Rev. D84 (2011) 054019, Applied in πN scattering on the level of invariant amplitudes PENALTY FUNCTION INTRODUCED 1.E. Pietarinen, Nuovo Cimento Soc. Ital. Fis. 12A, 522 (1972). 2.Hoehler – Landolt Boernstein „BIBLE” (1983)

17. PWA8/ATHOS3 201517 What is Pitarinens expansion? In principle, in mathematical language, it is ”...a conformal mapping which maps the physical sheet of the ω-plane onto the interior of the unit circle in the Z-plane...” In practice this means:

18. PWA8/ATHOS3 201518 Or in another words, Pietarinen functions Z(ω) are a complet set of functions for an arbitrary function F(ω) which HAS A BRANCH POINT AT x P ! Observe: Pietarinen functions do not form a complete set of functions for any function, but only for the function having a well defined branch point.

19. PWA8/ATHOS3 201519 Powes series for Z(ω) = Illustration:

20. PWA8/ATHOS3 201520 Z(ω)

21. PWA8/ATHOS3 201521 Z(ω) 2

22. PWA8/ATHOS3 201522 Z(ω) 3

23. PWA8/ATHOS3 201523 Utilization

24. PWA8/ATHOS3 201524 Collaboration RBI Zagreb Alfred Svarc (AS) Sasa Ceci (SC) UT Tuzla Jugoslav Stahov (JS) Hedim Osmanovic (HO) Mirza Hadzimehmedovic (MH) JGU Mainz Lothar Tiator (LT) Michael Ostrick (MO) Viktor Kashevarov (VK) Kiril Nikonov (KN) Sven Schumann (SS) GWU Washington DC Ronald W. Workman (RW)

25. PWA8/ATHOS3 201525 I. obtaining poles from SC ED and SE PWA partial wave solutions using L+P method (on the level of partial waves)

26. PWA8/ATHOS3 201526

27. PWA8/ATHOS3 201527 a. Published

28. PWA8/ATHOS3 201528 b. Work in progress 1. Introducing multichannel L+P formalism tested on πN → πN and πN → ηN P11 partial wave from Bonn-Gatchina BG2012-2 solution (AS,MH,HO,JS,LT,RW)

29. PWA8/ATHOS3 201529 2. Developing EtaMAID-2015 (VK, LT, MO, …, AS, MH, HO, JS)

30. PWA8/ATHOS3 201530 II.New SC SE PWA for γN → ηN process (new MAID) (on the level of invariant amplitudes) a’la Karlsruhe Helsinki πN el. PWA (AS, MH, HO, JS, LT, MO, VK. KN, SS)

31. PWA8/ATHOS3 201531 Stahov, MAID collaboration meeting, Mainz 2015

32. PWA8/ATHOS3 201532 Why?

33. PWA8/ATHOS3 201533 In 2014 new SE SC fixed - W PWAs for γN→ηN were developed (Independently by Mainz & UT-RBI) Problems: Ambiguities appear! Example:

34. PWA8/ATHOS3 201534 Explanation: CONTINUUM AMBIGUITIES

36. PWA8/ATHOS3 201536 Several ways of eliminating CA: Coupled channels (restores unitarity, shrinks „islands” to points, restores uniqueness)Coupled channels (restores unitarity, shrinks „islands” to points, restores uniqueness) Imposing analyticityImposing analyticity …….……. Several ways of eliminating CA: Coupled channels (restores unitarity, shrinks „islands” to points, restores uniqueness)Coupled channels (restores unitarity, shrinks „islands” to points, restores uniqueness) Imposing analyticityImposing analyticity …….…….

37. PWA8/ATHOS3 201537 Procedure: PWA fixed t Tiator, MAID collaboration meeting, Mainz 2015 PWA fixed W Step 1 fixed t PWA Step 2 fixed W PWA We start with (W,t) figure

38. PWA8/ATHOS3 201538 Stahov, MAID collaboration meeting, Mainz 2015

39. PWA8/ATHOS3 201539 The trick: Observables are given in terms if a combination of invariant amplitudes

40. PWA8/ATHOS3 201540 Stahov, MAID collaboration meeting, Mainz 2015

41. PWA8/ATHOS3 201541 Stahov, MAID collaboration meeting, Mainz 2015

42. PWA8/ATHOS3 201542 Stahov, MAID collaboration meeting, Mainz 2015

43. PWA8/ATHOS3 201543 Tricks and issues: Creating t-dependent data baseCreating t-dependent data base Initial minimization in tInitial minimization in t Initial minimization in WInitial minimization in W IterationIteration First testing (t-fitting) was successfully done on EtaMAID2015-b pseudo data experimental data Analyzing real data is IN PROGRESS

44. PWA8/ATHOS3 201544 Creating t-dependent data base Tiator, MAID collaboration meeting, Mainz 2015

45. PWA8/ATHOS3 201545 Kashevarov, MAID collaboration meeting, Mainz 2015 Fixed W

46. PWA8/ATHOS3 201546 Fixed t

47. PWA8/ATHOS3 201547 Fixed t

48. PWA8/ATHOS3 201548 Fixed t

49. PWA8/ATHOS3 201549