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Alfred Švarc Alfred Švarc Ruđer Bošković Institute Ruđer Bošković Institute

CroatiaCroatia

Bare propagator poles in coupled-channel Bare propagator poles in coupled-channel modelsmodels

(Possible link between microscopic theories and (Possible link between microscopic theories and phenomenological models)phenomenological models)

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The short history:

In last few years I have been faced with two problems

1. Are the off-shell effects measurable?2. How can we understand bare coupled-channel quantities?

I have asked these questions at few workshops and conferences, and

it turned out that these problems seem to be it turned out that these problems seem to be relatedrelated..

What is in common?

1. Both problems originate in an attempt to link microscopic to macroscopic effects

2. Both problems are controversial because basic field theoretical arguments “forbid” what seems to be very plausible on the macroscopic level

1. Can we formulate the problem here exactly?

2. Can we make a step forwards towards giving a competent answer to the existing controversy?

The question are:

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A brief summary of the off-shell problem

calculating processes with more then 2-nucleons requires an assumption about the off-shell behavior of the 2-body amplitude

it has been widely accepted that off-shell behavior is a measurable quantity (like for instance in Nucleon-Nucleon Bremsstrahlung, pion photo- and electro production or real and virtual Compton scattering on the nucleon)

many different models for the off-shell extrapolations have been suggested and the results compared

A controversy has arisen when Fearing and Scherer declared that the off-shell effects are unmeasurable because of first field-theoretical principles

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Maybe the answer lies in this part of conclusions?

My dilemma:

• we do need model off-shell form factors to calculate any observable in a more then 2-body process, and different models give different results

• if we can not establish the correctness of the off-shell form factors, that means that we in principle can not calculate anything at all

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A brief summary of the bare propagator problem

coupled-channel formalism has been known for decades, but (at least to my knowledge) no credible physical meaning to the bare quantities is given in spite of general agreement that bare quantities are obtained when self energy contributions are deducted (singled out, taken away)

the idea to relate bare quantities to the quark-model-calculation ones has appeared (references follow)

a controversy has arisen when the proposal has been criticized because of incompatibility with the first field-theory principles

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From now on I will present some facts related to the possible understanding of

bare quantities in coupled-channel models

I will restrict my discussion to bare propagator pole I will restrict my discussion to bare propagator pole values.values.

Why poles?Why poles?

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The formulation of hadron spectroscopy program

A. Švarc, 2ndPWA Workshop, Zagreb 2005

Höhler – Landolt Bernstein

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As nothing better has been offered quark model resonant states are up to now directly identified with the scattering matrix singularities obtained directly from the experiment.

Most single channel theories recognize only one type of scattering matrix singularity – scattering matrix pole.

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Up to now:

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However, coupled channel models, based on solving Dyson-Schwinger integral type equations having the general structure

full = bare + bare * interaction* full

do offer two types of singularities:• bare poles• dressed poles

Questions: 1. How do we extract bare and dressed propagator poles?

2. What kind of physical meaning can we assign to dressed and/or bare propagator poles?

0 0G G G G

According to my knowledge, no physical meaning to the bare propagator polesbare propagator poles

in the coupled-channel formalism has ever been given.

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Should be that done?

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A tempting possibility has been suggested in 1996. by Sato and Lee within the framework of dynamical coupled-channel model, and elaborated for photoproduction of Δ-resonance (γN → Δ):

quark-model quantities cc-model bare value

quantities

Question: Can the idea be justified?

Details given in

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The idea has been repeated since:

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2004

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The controversy exists!

Strong criticism of such an idea has been made by C. Hanhart and S. Sibirtsev at ETA07 in

Peniscola

The criticism is based on incompatibility of such an interpretation with some first principles originating in the

field theory.

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I will now give a short preview of the essential from:

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So, in such a type of a model (as in any coupled-channel model) we have two type of quantities: bare and dressed ones

bare:

bare vertex interaction

bare resonant state masses

dressed: dressed vertex interaction

dressed resonant state masses

defined by equation

defined by equation

(when dressed propagator in resonant contribution is diagonalized)

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UP TO NOW quark model resonant states

scattering matrix poles

Problems for transition amplitudes

Proposed way out

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Applied to Δ → γN helicity amplitudes

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Extension to the full N* resonance spectra is proposed in Matsuyama, Sato and Lee, Physics Reports 439 (2007):

However it is not yet done:

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So, let me give a short resume:

1. At our disposal we have two kind of singularities to be discussed: bare and dressed scattering amplitude poles.

2. The speculation to identify bare quantities in a cc model with quark model ones is introduced

3. The idea has not been proven (controversy in interpretation exists)

4. The idea is verified for γN →Δ helicity amplitudes obtained when using bare and dressed interaction vertices, and the good agreement is found

5. The necessity to extent it to the full N* spectrum is stressed

6. No systematic results for coupled-channel models are given yet, but preliminary reports from a number of groups do exist

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A comparison of bare propagator poles with constituent quark-model predictions

is for the whole N* spectrum given for a:

coupled-channel model of CMB type where the interaction is effectively represented with an entirely

phenomenological term.

Let me give an example of one:

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Carnagie-Melon-Berkely (CMB) model

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.

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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

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0 0G G G G

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

where

we obtain the final expression

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What should be identified with what?

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bare propagator pole position mass of a quark-model resonant state

imaginary part of the dressed propagator pole

decay width

Following the idea from photoproduction:

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What is our aim?

To establish if there is any regular pattern of behavior .

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Results

Model:

1. CMB model with three channels

πN, ηN and π2 N - effective 2-body channel

2. Input:

πN elastic: VPI/GWU single energy solution πN → ηN: Zagreb 1998 PWA data

3. Quark model quantities are taken from Capstick-Roberts constituent quark model

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The intention is to ask for the absolute minimum!

To see if the interpretation of bare propagator poles as quark-model resonant state is allowed for the used input data set.

We perform a constrained fit with the bare propagator pole values fixed to the quark-model values!

Of course, we shall investigate whether the solution is:

• unique

• best

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The comparison is done for lowest partial waves

S11 , P11 , P13 and D13

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Let us show the two lowest parity odd states:

11 1331 and D2 2S

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2 2.6R

2 3.4R

S11 12

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S11 12

dressed pole

quark model resonant state

constrained fit bare propagator mass

free fit bare propagator mass

PDG

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S11 12

dressed pole

quark model resonant state

constrained fit bare propagator mass

free fit bare propagator mass

PDG

1.559 1.727 1.803 2.090

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2 2.0R

2 1.5R

πN elastic πN → ηN

D13 32

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D13 32

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1.590 1.753 1.972 2.162

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Let us show the two lowest parity even states states:

11 1331 and P2 2P

Problems appear

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πN elastic πN → ηN

P13 32

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P13 32

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1.725 1.922 2.220

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πN elastic πN → ηN

P11 NOTORIOUSLY PROBLEMATIC ONE 12

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P11 12

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1.612 1.728 2196

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Conclusions:

1. There is a certain level of resemblance between bare propagator poles in a CMB type coupled-channel model and constituent quark model resonant states

2. There is a certain level of resemblence between our bare propagator poles and Mainz group results.

3. The mechanism is established to distinguish between genuine scattering matrix pole – generated by a nearby bare propagator pole and a dynamic scattering matrix pole which is generated by the interference effect among distant bare propagator poles

4. The Roper resonance is in this model consistent with being a dynamic scattering matrix pole

5. New partial wave data from other inelastic channels are required in order to further constrain the fit, and give a more confident answer about the precise position and nature of a scattering matrix resonant state under observation

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Final question to be answered here:

What is the correspondence between bare propagator poles in general and hadron structure calculations?