Partial Wave and N* Analysis with Analyticity

Alfred SvarcRudjer Boskovic Institute, Zagreb,

Croatia

 

PWA8/ATHOS3 2015 1

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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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Current (PDG) definitionResonance … scattering matrix pole

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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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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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Analyticity in ED PWA

Usually implemented through analyticity of S-matrix functions.

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

?

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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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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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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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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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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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Utilization

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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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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)

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

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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

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

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

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

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

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

PWA8/ATHOS3 2015 45Kashevarov, MAID collaboration meeting, Mainz 2015

Fixed W

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Fixed t

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Fixed t

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Fixed t

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Thank you for your patience