Partial Wave and N* Analysis with Analyticity Alfred Svarc Rudjer Boskovic Institute, Zagreb,...
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![Page 1: Partial Wave and N* Analysis with Analyticity Alfred Svarc Rudjer Boskovic Institute, Zagreb, Croatia PWA8/ATHOS3 20151.](https://reader036.fdocument.org/reader036/viewer/2022062516/56649d945503460f94a7bc52/html5/thumbnails/1.jpg)
Partial Wave and N* Analysis with Analyticity
Alfred SvarcRudjer Boskovic Institute, Zagreb,
Croatia
PWA8/ATHOS3 2015 1
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PWA8/ATHOS3 2015 2
What is resonance?This is an old question…..
Possible answers:• backwards looping of Argand diagram• existence of time delay• scattering phase going through π/2• structure in speed plot• pole in the partial wave scattering matrix
……………….
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PWA8/ATHOS3 2015 3
Current (PDG) definitionResonance … scattering matrix pole
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PWA8/ATHOS3 2015 4
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PWA8/ATHOS3 2015 5
SINGLE ENERGY (SE)
ENERGY DEPENDENT (ED)
SIN
GL
E C
HA
NN
EL
(SC
)C
OU
PL
ED
CH
AN
NE
L
(C
C)
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: AMBIGUITIESSolution: imposing A and U
constraints
Very little has been done
What kind of PWA do we have?
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PWA8/ATHOS3 2015 6
Analyticity
Scattering matrix is an analytic function in Mandelstam s, t and u variables.
Scattering matrix is an analytic function in Mandelstam s, t and u variables.
Why do we need analyticity?
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PWA8/ATHOS3 2015 7
Analyticity in ED PWA
Usually implemented through analyticity of S-matrix functions.
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PWA8/ATHOS3 2015 8
Analyticity in SE PWA
Problem: How to implement analyticity because SE PWA is amplitude reconstruction in a discrete
set of energy points?
• analytic penalty function
What is the choice of analytic penalty function?
• theoretical models • Pietarinen expansion
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Analyticity in s (W)(fixed W; usually exploited)
Tiator, MAID collaboration meeting, Mainz 2015
PWA fixed W
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PWA8/ATHOS3 2015 10
Analyticity in t = f(W,θ)(fixed t; very rarely mentioned!)
Tiator, MAID collaboration meeting, Mainz 2015
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Analyticity in t = f(W,θ)
Tiator, MAID collaboration meeting, Mainz 2015
PWA fixed t
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PWA8/ATHOS3 2015 12
We still didn’t say how we introduce analyticity constraints!
?
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PWA8/ATHOS3 2015 13
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!
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PWA8/ATHOS3 2015 14
Pietarinen expansion
• SE SC PWA
• Extraction of poles from SE and ED partial waves
New method (2013)
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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
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PWA8/ATHOS3 2015 16
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 αs", Phys.Rev. D84 (2011) 054019,
Applied in πN scatteringon 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)
NAMING !
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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:
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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 xP !
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.
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Powes series for Z(ω) =
Illustration:
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Z(ω)
4 2 2 4s
0.2
0.4
0.6
0.8
1.0
Real
4 2 2 4s
0.8
0.6
0.4
0.2
Imag
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PWA8/ATHOS3 2015 21
Z(ω)2
4 2 2 4s
0.2
0.2
0.4
0.6
0.8
1.0
Real
4 2 2 4s
1.0
0.8
0.6
0.4
0.2
Imag
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PWA8/ATHOS3 2015 22
Z(ω)3
4 2 2 4s
1.0
0.5
0.5
1.0
Real
4 2 2 4s
1.0
0.8
0.6
0.4
0.2
Imag
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PWA8/ATHOS3 2015 23
Utilization
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PWA8/ATHOS3 2015 24
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)
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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/ATHOS3 2015 26
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a. Published
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b. Work in progress1. 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)
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2. Developing EtaMAID-2015 (VK, LT, MO, …, AS, MH, HO, JS)
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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)
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PWA8/ATHOS3 2015 31Stahov, MAID collaboration meeting, Mainz 2015
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PWA8/ATHOS3 2015 32
Why?
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In 2014 new SE SC fixed - W PWAs for γN→ηN were developed(Independently by Mainz & UT-RBI)
Problems: Ambiguities appear!
Example:
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Explanation: CONTINUUM AMBIGUITIES
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Several ways of eliminating CA:
• Coupled channels (restores unitarity, shrinks „islands” to points, restores uniqueness)
• Imposing analyticity• …….
Several ways of eliminating CA:
• Coupled channels (restores unitarity, shrinks „islands” to points, restores uniqueness)
• Imposing analyticity• …….
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Procedure:
PWA fixed t
Tiator, MAID collaboration meeting, Mainz 2015
PWAfixed W
Step 1 fixed t PWAStep 2 fixed W PWA
We start with (W,t) figure
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PWA8/ATHOS3 2015 38Stahov, MAID collaboration meeting, Mainz 2015
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PWA8/ATHOS3 2015 39
The trick:Observables are given in terms if a combination of invariant amplitudes
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PWA8/ATHOS3 2015 40Stahov, MAID collaboration meeting, Mainz 2015
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PWA8/ATHOS3 2015 41Stahov, MAID collaboration meeting, Mainz 2015
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PWA8/ATHOS3 2015 42Stahov, MAID collaboration meeting, Mainz 2015
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Tricks and issues:
• Creating t-dependent data base• Initial minimization in t• Initial minimization in W• Iteration
First testing (t-fitting) was successfully done onEtaMAID2015-b pseudo data experimental data
Analyzing real data is IN PROGRESS
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Creating t-dependent data base
Tiator, MAID collaboration meeting, Mainz 2015
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PWA8/ATHOS3 2015 45Kashevarov, MAID collaboration meeting, Mainz 2015
Fixed W
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Fixed t
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PWA8/ATHOS3 2015 47
Fixed t
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PWA8/ATHOS3 2015 48
Fixed t
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PWA8/ATHOS3 2015 49
Thank you for your patience