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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
88
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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:
11111111
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?
1313
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:
1515
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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
3737
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
4040
D13 32
4141
1.590 1.753 1.972 2.162
4242
Let us show the two lowest parity even states states:
11 1331 and P2 2P
Problems appear
4343
πN elastic πN → ηN
P13 32
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P13 32
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1.725 1.922 2.220
4646
πN elastic πN → ηN
P11 NOTORIOUSLY PROBLEMATIC ONE 12
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P11 12
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1.612 1.728 2196
4949
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
5050
Final question to be answered here:
What is the correspondence between bare propagator poles in general and hadron structure calculations?