Inherent model dependence of Breit-Wigner parameters

(poles as a true signal of resonance properties)

A. Švarc

Rudjer Bošković Institute, Zagreb, Croatia

INT-09-3

The Jefferson Laboratory Upgrade to 12 GeV

(Thursday, November 12, 2009)

Batinic 95

Publications

Breit-Wigner parameters and poles are discussed within the scope of PWA - hadron spectroscopy

What is hadron spectroscopy ?

What is the aim of hadron spectroscopy?

Höhler – Landolt Bernstein 1984.

What we actually do and who is doing what?

BEFORE NOWADAYS

And how are the results currently presented?

Breit-Wigner parameters Pole positions

PDG2008

What do we know about Breit-Wigner parameters?

What do we know about poles?

We have to ask ourselves two questions:

We know for decades:Breit-Wigner parameters are MODEL DEPENDENT!

continuum ambiguities

What do we know about Breit-Wigner parameters?

But UNFORTUNATELY this fact is not unanimously accepted!

A simple iIllustration of BW model dependence:

And what is the way out?

Use another, more model independent quantity:as recommended in PDG: POLES

Can we do any recommendations here?

Before this morning I wanted to suggest:

• Establish a consensus if this is really true• Write a note in the “white paper”• Ask PDG (C. Wohl) to restore the introductory part about

Breit-Wigner model dependence• Suggest to PDG to change the order of appearance, to put

the poles first• Write a paper in which one should show if there is any

difference if pole positions are used instead of BW parameters, and if not where to expect the effect

But now I believe that we should

There is no fundamental difference between a bound state and a resonance, other than the matter of stability, it is to be expected that when simple poles of the coupled channel amplitude occur on unphysical sheets in the complex energy plane, they ARE TO BE associated with resonant states.

Mandelstam hypothesis

What do we know about poles?

What do we measure? Quantities on the physical axis

What are we looking for? Poles in the complex energy plane

Problem: How do we parameterize simple poles in the complex energy plane when we can only see data lying on the physical axis?

Let us see what our problem is

Our conclusion:

a) Breit-Wigner parameters : model dependent

their interpretation is not clear recommendation:

do not extract without defining the procedure

b) Poles are recommended

However in QCD ....

Questions appear:

1. How many type of poles do we have?2. Are some of them “more fundamental” then the others?3. Are all of them the “image” of some internal structure ?

As physical observable can be expressed in terms of either T or K matrices, from ancient times we know that we have:

1. T – matrix poles2. K – matrix poles and from recently we know that we have3. Bare poles

There are many ways of looking for the poles, but I would like to show one of them we are able to apply

Methods

Breit-Wigner parameters

Speed-Plot

• constant background• energy dependent background• Flatte

Time-delay

N/D method

Analytic continuation

Regularization method

The procedure to analyze the T matrix poles we present now was first proposed at NSTAR2005 in Tallahassee and

elaborated at the BRAG2007 meeting in Bonn:

As extracting poles is related to the exact form of the analytic function which is used to describe PW amplitudes, we propose to use only one method to extract pole positions from published partial wave analyses understanding them as nothing else but good, energy dependent representations of all analyzed experimental data – as partial wave data (PWD).

We have chosen the 3-channel T-matrix Carnegie-Melon-Berkeley (CMB) model for which we have developed our own set of codes and fitted “ALL AVAILABLE” PWD or PWA we could find “on the market”.

What is the benefit?

1. Alll errors due to different analytic continuations of different models are avoided, and the only remaining error is the precision of CMB method itself.

2. Importance of inelastic channels IS CLEARLY VISIBLE

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

0 0G G G G

CMB coupled-channel model

0 0 0 0 0G G G G G G G

Carnagie-Melon-Berkely (CMB) model is an isobar 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.

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

channel-resonance mixing matrix

bare particle propagatorchannel propagator

Assumption: The imaginary part of the channel propagator is defined as:

2 2( ( ) )( ( ) )( )

4a

s M m s M mq s

s

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

And we require its analyticity through the dispersion relation:

0 0G G G G

we obtain the full propagator G by solving Dyson-Schwinger equation

where

we obtain the final expression

Input data

π-N elastic, Imaginary part

The analysis is done for the S11 partial wave.

Analysis of other partial wave is in progress.

π-N elastic, Real part

πN elastic channel:

Gross features of ALL partial wave amplitudes are the same• “two peak structure” is always present

Minor differences in imaginary part:• Peak position is sometimes slightly shifted (second peak for Giessen)• Size of the peaks is different

• Second peak is much narrower for Giessen• First peak is somewhat lower for Giessen• Second peak is much lower for EBAC

• The structure of higher energy part is different (smooth for FA08, much more structure for others)

πN → ηN, Imaginary part

πN → ηN, Real part

πN →ηN channel:

Gross features of ALL partial wave amplitudes are similar• “two peak structure” is visible

Notable differences in imaginary part:• Peak strength is much higher in EBAC• Second peak starts very early for Giessen, somewhat later for EBAC

and much later for Zagreb • Strength of the higher energy peak is shifted to the real part for

Zagreb

Present situation with poles.

Poles in PDG2008:

Our solutions:

We have two sets of solutions:

1. We have fitted πN elastic channel only2. We have fitted πN elastic + πN→ηN

πN elastic, imaginary part

Typical results:

Private communication V. Skhlyar

Private communication V. Skhlyar

Private communication J. Diaz

Private communication J. Diaz

πN elastic

πN elastic + πN → ηN

πN elastic

πN elastic + πN → ηN

Problems with EBAC

It seems that in EBAC model it is impossible to simultaneously fit πN elastic and πN→ηN data because one has to significantly modify elastic channel in order to fit them both!

Explanations

1. Sign problem of some of one of the effective Lagrangian contributions

2. Some problems with the threshold opening

3. Too few resonances? (EBAC has two, Mainz 4, Zagreb 3 – 4)

4. Numeric error?

Problem:

CMB/LBL model, as the word says, is MODEL dependent

How big is the model dependence?

If we want to analyze the behavior of poles of the “world collection” of PWD and PWA we have to answer the question:

Idea:

1. We take a well defined input for which the T-matrix poles are extremely well defined

2. Modify the ingredients of the model

3. Check the stability of the solution

Realization

We take the Zagreb amplitudes for which the poles are well known and stable for THREE CHANNELS

πN→πN

πN→ηN

πN→π2N

1. Input

Assumption: The imaginary part of the channel propagator is defined as:

2. What is model dependent ?

a. Channel propagator

b. Number of channels

c. Parameterization of the background contribution

1. We modify the channel vertex function Im a

2. We perform the fit

a. Channel propagator

Results are PRELIMINARY, pole positions are fairly independent with respect to model ingredients.

Conclusions - I

1. Breit – Wigner parameters are inherently model dependent

2. One should concentrate on pole positions and residues

Conclusions - II

1. All PWA input data are similar, but not identical

2. We used only one method to extract resonance parameters from “all available” PWA understanding them as PWD in order to determine how much the difference in appearance of input means the difference in resonance positions

3. We have chosen to compare poles positions as indicators of resonance properties

4. The number of poles turns out to be the “issue of relevance” even before we start the fit

5. The size of reduced χ2 indicates• The level of analyticity of analyzed PWA• The similarity of the chosen functional form

with the CMB model

8. If the second channel is added, the need for the 3rd pole is confirmed, but the results are still not conclusive. Adding the third pole “reduces” the ambiguity in the third pole position, but shifts the position of the first two poles.

WE NEED MORE CONSTRAINTS

6. Two poles are unambiguously determined in ALL PWA if we fit the elastic channel only (1535 and 1650). Their averaged value is extracted.

7. The fit to elastic channel shows only some improvement if the third pole is added, but there is no conclusive proof that we do need one.

The third pole is loosely defined, and we have several, almost equivalent solutions (either higher in energy and close to the physical axes or closer in energy but far in the complex energy plane)

WE NEED ADDITIONAL CONSTRAINTS

11. Importance of polarization measurements:We hope to get independent constraints on real and imaginary part of the partial wave T-matrix, and not only on the absolute value.

10. Conclusion: to constrain ALL poles we need ALL channels

Corollary 1: measuring only one channel precisely IS NOT ENOUGH to fix all poles.

9. The new S11(1846) state seen in photo-production channel (Mainz-Taipei) is consistent with all published PWA.

Corollary 2: As the coupling of baryon resonances to the electromagnetic channel is rather weak (electromagnetic channels are practically decoupled from the strong ones), one should be very cunning to use the photopruduction measurements in order to reduce the ambiguity in standard baryon resonance pole positions.