Partial Wave and N* Analysis with Analyticity Alfred Svarc Rudjer Boskovic Institute, Zagreb, Croatia PWA8/ATHOS

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 ……………….

PWA8/ATHOS Current (PDG) definition

PWA8/ATHOS

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?

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

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

PWA8/ATHOS 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

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

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

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

PWA8/ATHOS We still didn’t say how we introduce analyticity constraints!

PWA8/ATHOS 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!

PWA8/ATHOS 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)

PWA8/ATHOS 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

PWA8/ATHOS 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) , 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)

PWA8/ATHOS 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:

PWA8/ATHOS 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.

PWA8/ATHOS Powes series for Z(ω) = Illustration:

PWA8/ATHOS Z(ω)

PWA8/ATHOS Z(ω) 2

PWA8/ATHOS Z(ω) 3

PWA8/ATHOS Utilization

PWA8/ATHOS 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)

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

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PWA8/ATHOS a. Published

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

PWA8/ATHOS Developing EtaMAID-2015 (VK, LT, MO, …, AS, MH, HO, JS)

PWA8/ATHOS 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)

PWA8/ATHOS Stahov, MAID collaboration meeting, Mainz 2015

PWA8/ATHOS Why?

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

PWA8/ATHOS Explanation: CONTINUUM AMBIGUITIES

PWA8/ATHOS 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 …….…….

PWA8/ATHOS 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

PWA8/ATHOS Stahov, MAID collaboration meeting, Mainz 2015

PWA8/ATHOS The trick: Observables are given in terms if a combination of invariant amplitudes

PWA8/ATHOS Stahov, MAID collaboration meeting, Mainz 2015

PWA8/ATHOS Stahov, MAID collaboration meeting, Mainz 2015

PWA8/ATHOS Stahov, MAID collaboration meeting, Mainz 2015

PWA8/ATHOS 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

PWA8/ATHOS Creating t-dependent data base Tiator, MAID collaboration meeting, Mainz 2015

PWA8/ATHOS Kashevarov, MAID collaboration meeting, Mainz 2015 Fixed W

PWA8/ATHOS Fixed t

PWA8/ATHOS Fixed t

PWA8/ATHOS Fixed t

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