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Poles of PWD ata and PW A mplitudes in Zagreb model. A. Švarc , S. Ceci, B. Zauner Rudjer Bošković Institute, Zagreb, Croatia M. Hadžimehmedović , H. Osmanović , J. Stahov Univerzity of Tuzla, Tuzla Bosn ia and Herzegovina. How do I see what is our main aim?.

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poles of pwd ata and pw a mplitudes in zagreb model

Poles of PWDataand PWAmplitudesin Zagreb model

A. Švarc, S. Ceci, B. Zauner

Rudjer Bošković Institute, Zagreb, Croatia

M. Hadžimehmedović, H. Osmanović, J. Stahov Univerzityof Tuzla, TuzlaBosniaandHerzegovina

slide2
How do I see what is our main aim?

Experiment

Quarks

Matching point

?

bound states

structures

resonances

slide3
Höhler – LandoltBernstein 1984.

Burkert – Lee 2004.

Höhler – LandoltBernstein 1984.

Svarc 2004

Ceci, Svarc, Zauner 2005.

slide4
Difference between PWD and PWA

Höhler – LandoltBernstein 1984.

explicit analytic form introduced

  • Phenomenological T-matrix
    • CMU-LBL
    • Zagreb
    • Argonne-Pittsburgh
  • effective Lagrangian
    • EBAC
    • Juelich
    • Dubna-Mainz-Taipei (DMT)
    • Giessen
  • Chew-Mandelstam K-matrix
    • GWU/VPI

PWD

PWA

slide5
How do I see what is our main aim?

Experiment

Quarks

Breit-Wigner parameters

bound states

Pole parameters

structures

resonances

Phase shifts

slide6
What is “better”:
    • Breit-Wigner parameters
    • or
    • Pole parameters

The advantages and drawbacks

slide7
Breit – Wigner parameters:
  • Advantages:
    • defined on the real axes
    • simple to calculate
  • Drawbacks:
    • dependence on the choice of field variables
    • model dependent (background definition)

Harry Lee BRAG 2001

slide11
Pole parameters
  • Advantages
    • invariant with respect to the choice of field variables
    • model independent
    • less model dependent
  • Drawbacks
  • hard to get because they lie in the complex energy plane

Why?

slide12
Extractionof Breit-Wignerparameters

The dependence upon background parameterization is a well know fact, but nothing has been done in PDG yet.

PDG makes an average of all BW values, regardless of the way background has been introduced. This introduces an additional systematic error.

slide13
Suggestion by Harry Lee:

Suggestion by Lothar Tiator:

We all know that BW positions and parameters are not well defined but many people believe that within some uncertainty they can be given, and are very useful. Therefore I also tend to stick with them and would propose to keep them in PDG in the future. But only if we can give some proper definition and methods for extraction.

slide14
Extractionof pole parameters
  • Each PWA has a specific assumption on the analytic form.
  • Idea:
  • Let us use CMB formalism to analyze available PWD and PWAusingone andthe same analyticform.
  • We propose:
    • To use new PWD or PWA in addition to the existing ones and look for the shift of poles (shift of present ones, appearance of the new ones)
    • To use ONE analytic form (Zagreb CMB) for ALL EXISTING PWA and PWD, and:
      • extract poles from a particular PWD or PWA using Zagreb CMB and compare the outcome with the original result
      • compare agreement of poles of ALL EXISTING PWA and PWD in order to avoid systematic error because of differences in analytic forms.
slide15
Warning:

When we analyze particularPWA, we do not say that we shall exactly reproduce pole positions given by thatparticular model.

Results may differ, and the difference will show the significance of analytic form chosen to represent on-energy shell data.

So, we are not checking if the pole positions in a certain model are correctly extracted, but rather seting up the way how to quantify the comparison of different curves.

slide17
In details repeated at NSTAR2005 - Tallahassee

Technical problems on Zagreb side......

slide18
Using CMB to analyze a world collection of PWD and PWA and eliminate model assumptions on analytic form
  • Formulated at BRAG2007
  • Research in progress

REMARK

Technical problems in Zagreb code are now eliminated. Code is running under LINUX. It is transferable from machine to machine, and is an open source code. Adjustments and improvements can be done.

(I can demonstrated how the code works during workshop)

slide19
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
slide20
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.

slide21
We represent the full T-matrix in the form where the channel-resonance interaction is not calculated but effectively parameterized.

Model is manifestly unitary and analytic.

bare particle propagator

channel-resonance mixing matrix

channel propagator

slide22
Model assumptions:
      • isobar model (poles are introduced as intermediate resonant states called intermediate particles)
      • background parameterization - meromorphic function
      • the form of imaginary part of the channel propagator introduces proper channel cuts

Imaginary part of the channel propagator is defined as:

where qa(s) is the meson-nucleon cms momentum:

slide23
The analyticity is manifestly imposed by calculating the channel propagator real part through the dispersion relation:

qa(s) is the meson-nucleon cms momentum:

The unitarity has been proven by Cutkosky.

slide25
Step 1: Fitting procedure
  • we define the number of background poles
  • we define the number of resonance poles
  • we fit
  • si......... resonance mass
  • ic ......... channel resonance mixing parameters

Final result: energy dependent partial wave T-matrices on the real axes

Step 2: Extracting resonance parameters – go into the complex energy plane (singularity structure of the obtained solution)

slide27
To find the position of poles of the matrix T(s) in the complex energy plane we have to find all zeroes of the denominator. So, we solve the following equation:
slide28
How do we solve it?
    • when obtained from the fit, det G-1 (s)is a complex function of a real argument s
    • we have to
      • analytically continue this function into the complex energy plane (observe that only channel propagator (s) has to be analytically continued
      • find a complex zero s0 of that function in the complex energy plane – we do it numerically
slide29
Analytical continuation of the channel propagator (s)
    • Numerical integration (In old paper)
    • Nowadays - Pietarinen expansion

We have constructed a function:

Observe that this is a complex

function of a complex argument

for physical argument x!

for x > x0 x0 – x is negative, and ZI (x) is complex

slide32
Finding a complex zero
  • We did it numerically:
    • instead of calculating | det G -1| we have calculated | det G |
    • we made a 3D plot
    • | det G | = f (Re s, Im s)
    • and numerically looked for the
    • pointof infinity of this function.
slide35
Mass →

Width →

Partial width →

slide36
Stability of the procedure
  • Stability of the procedure has been tested with respect with different model assumptions of Zagreb CMB:
    • Defining the input
    • Form of the channel propagator (meson-resonance vertex function)
      • Inner part
      • Asymptotic part
      • Cut off parameters
    • Type of the background
    • Number of channels
    • Mass of the effective channel

(To be given at the end of the talk if time permits....)

results

Results

Use Zagreb CMB fits to analyze a particular PWD or PWA

Compare all PWA and PWD in Zagreb CMB in order to avoid systematic differences in analytic continuation

slide38
Importance of inelastic channels

Elastic channels only are insufficient to constrain all T-matrix poles, especially those which dominantly couple to inelastic channels.

slide40
We use:
  • CMB model for 3 channels:
  • p N, h N, and dummy channel p2N
  • p N elastic T matrices , PDG: SES Ar06
  • p N¨h N T matrices, PDG:Batinic 95

We fit:

πNelastic only

p N¨h N only

both channels

slide45
Use Zagreb CMB fits to analyze a particular PWD or PWA

I. Dubna – Mainz – Taipei (DMT)

S11 : DMT model fits GWU/VPI single energy solutions , and obtains:

slide46
TheyalsofitπN→ηN S11

But we shall return to importance of elastic channels later.

slide47
Dilemma: How many dressed poles does one find in DMT functions?
  • Facts of life:
      • DMT model has 4 bare poles in S11
      • bare poles gets dressed, travel from the real axes into the complex energy plane, and there is no a priori way to say where they end
      • one needs either analytic continuation of DMT functions or some other pole search method
      • up to now DMT uses speed plot technique
slide50
3R

4R

slide51
This is a good place to illustrate the difference in analytic structure between CMB and other models.

At the same time this is a good place to illustrate the model dependence of bare parameters.

Both, Zagreb and DMT have the same structure

full = bare + bare * interaction* full

and this brings us to the self energy, and theseparation to the bare and dressed quantities.

I will illustrate that the wayofseparation

bare ↔dressed quantities

is different in DMT and Zagreb.

slide52
I take DMT amplitudes, fit them with Zagreb CMB,but fix bare Zagreb masses to DMT values:

M10= 1559 MeV

M20= 1727 MeV

M30= 1803 MeV

M40= 2090 MeV

slide54
M10= 1559 MeV

S1 = 1600 + i 105

S2 = 1680 + i 750

M20= 1727 MeV

M30= 1803 MeV

S3 = 1790 + i 105

S4 = 2140 + i 350

M40= 2090 MeV

Χ2red = 0.63

Best fit

M10= 1506 MeV

S1 = 1501 + i 118

M20= 1650 MeV

S2 = 1640 + i 124

M30= 1853 MeV

S3 = 1921 + i 176

M40= 2118 MeV

S4 = 2129 + i 224

Χ2red = 0.07

slide55
Conclusion:

We support 4 bare pole solution, but find the 4th dressed pole too.

We see three poles under 2 GeV and one above.

The latest DMT analysis goes beyond speed plot, uses analytic continuation, and says something different: analytic continuation sees the pole where speed plot does not!

slide57
Compare Zagreb CMB fits with particular PWD or PWA

II. EBAC

Original motivation: strong change in πN elastic PW when the ηNdata are included

Diaz07 (πN elastic) PWA

Durand08 (πN elastic + πN → ηN) PWA

slide58
NormalizationofηN PW?

In publicationwefind

Guided by:

slide59
Wehaveanalysed
      • Diaz07 (πN elastic) PWA
      • Durand08 (πN elastic + πN → ηN) PWA
        • (as in publication)
      • Durand08 (πN elastic + πN → ηN) PWA (renormalized as suggestedby Dűring)
slide65
Renormalizatio suggested by Michael Dűring (December 2009)

Phasespacefactor

He replacedthe PS factor for πN elastic channel

withthe PS factor for πN → ηN channel

slide73
Compare all PWA and PWD in Zagreb CMB in order to avoid systematic differences in analytic continuation
slide88
Stability of the procedure
  • Stability of the procedure has been tested with respect with different model assumptions of Zagreb CMB:
    • Defining the input
    • Form of the channel propagator (meson-resonance vertex function)
      • Inner part
      • Asymptotic part
      • Cut off parameters
    • Type of the background
    • Number of channels
    • Mass of the effective channel
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