Alfred

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Bare propagator poles in coupled-channel models
(Possible link between microscopic theories and
phenomenological models)
Alfred Švarc
Ruder Boškovic
Institute Croatia
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• The short history
• In last few years I have been faced with two
  problems
• Are the off-shell effects measurable?
• 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 related.
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

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The question are

• Can we formulate the problem here exactly?
• Can we make a step forwards towards giving a
  competent answer to the existing controversy?

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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 values.
Why poles?
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The formulation of hadron spectroscopy program
Höhler Landolt Bernstein
A. Švarc, 2ndPWA Workshop, Zagreb 2005
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Most single channel theories recognize only one
type of scattering matrix singularity
scattering matrix pole.
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.
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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
• How do we extract bare and dressed propagator
  poles?
• What kind of physical meaning can we assign to
  dressed and/or bare propagator poles?

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According to my knowledge,
no physical meaning to the bare
propagator poles in the coupled-channel
formalism has
ever been given.
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 ?-resonance (?N ? ?)

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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(No Transcript)
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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
defined by equation
dressed resonant state masses
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 ? ? ?N 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
• At our disposal we have two kind of singularities
  to be discussed bare and dressed scattering
  amplitude poles.
• The speculation to identify bare quantities in a
  cc model with quark model ones is introduced
• The idea has not been proven
  (controversy in interpretation exists)
• The idea is verified for ?N ?? helicity
  amplitudes obtained when using bare and dressed
  interaction vertices, and the good agreement is
  found
• The necessity to extent it to the full N
  spectrum is stressed
• No systematic results for coupled-channel models
  are given yet, but preliminary reports from a
  number of groups do exist

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Let me give an example of one
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.
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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
bare particle propagator
channel-resonance
mixing matrix

channel propagator
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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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Following the idea from photoproduction
bare propagator pole position
mass of a quark-model resonant state
imaginary part of the dressed propagator pole
decay width
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What is our aim?
To establish if there is any regular pattern of
behavior .
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What do we expect?

• The PWD input should be reproduced
• The number of dressed resonances should
  correspond to PDG
• The number of bare propagator pole should
  correspond to the number of QMRS
• The grouping of bare propagator poles should
  correspond to the grouping of QMRS
• Each dressed propagator pole is generated by one
  bare propagator pole
• What did we get?
• The mechanism how to understand the missing
  resonance problem is offered
• One can visualize how the bare states get
  dressed
  (depict the travel from world without interaction
  into the real one)
• Identify whether the dressed state is generated
  by a single bare state, or in another more
  complicated manner (interference effect of
  distant poles)

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Results

• Model
• CMB model with three channels
• pN, ?N and p2 N - effective 2-body channel
• Input
• pN elastic VPI/GWU single energy solution
  pN ? ?N Zagreb 1998 PWA data
• 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

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S11
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S11
dressed pole
PDG
quark model resonant state
constrained fit bare propagator mass
free fit bare propagator mass
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S11
dressed pole
PDG
quark model resonant state
constrained fit bare propagator mass
free fit bare propagator mass
1.559 1.727 1.803 2.090
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D13
pN elastic pN ? ?N
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D13
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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
Problems appear
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P13
pN elastic pN ? ?N
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P13
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1.725 1.922 2.220
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P11
NOTORIOUSLY PROBLEMATIC ONE
pN elastic pN ? ?N
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P11
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1.612 1.728 2196
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• Conclusions
• There is a certain level of resemblance between
  bare propagator poles in a CMB type
  coupled-channel model and constituent quark model
  resonant states
• There is a certain level of resemblence between
  our bare propagator poles and Mainz group
  results.
• 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
• The Roper resonance is in this model consistent
  with being a dynamic scattering matrix pole
• 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?