Singularity structure of the nucleon

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Singularity structure of the ?-nucleon
scattering matrix
Alfred varc
Saa Ceci, Branimir
Zauner Ruder Bokovic Institute Croatia
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• Last year in Jülich
• an attempt to present ?-nucleon amplitudes in a
  form usable in ?-nucleus physics

(?)

• This year in Peniscola
• to summarize the last-year developments related
  to two body input

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Why are scattering matrix singularities important?

• What do we do in reality?
• We measure experimental data
• We perform
• partial wave analysis (PWA)
• Result
• partial wave data
• Resonances what are they ?

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The formulation of hadron spectroscopy program
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Formal definition Ceci, varc, Zauner, PRL 97,
062002 (2006) A central task of baryon
spectroscopy is to establish a connection between
resonant states predicted by various low energy
QCD models and hadron scattering observables. A
reasonable way to proceed seems to be to identify
poles of analytic scattering amplitudes which
simultaneously describe all experimental data in
all attainable channels with theoretically
predicted resonant states.
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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

Question
which
singularities are to be related to quark-model
resonant states?
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General structure of coupled-channel models
T.S.H. Lee

Models for extracting N
parameters from meson baryon reactions. Internati
onal Workshop on the Physics of Excited Baryons
(NSTAR 05), Tallahassee, Florida, 10-15 Oct
2005.
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• Let me just mention some of them
• KSU (Kent State University)
• VPI-GWU model (SAID)
• CMB model (Pittsburg-ANL, Zagreb)
• Unitary isobar model (UIM) Mainz group
• Giessen model
• KVI model (Gröningen)
• dynamical model (EBAC )

Main characteristics 1 - 3
phenomenological treatment of reaction mechanism
4 -7 reasonably well understood
reaction mechanism
Common denominator of all models
the integral structure of equations
is the same
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Up to now
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A new possibility has been suggested in dynamical
coupled-channel model, and elaborated for
photoproduction of ?-resonance (?N ? ?)
quark-model quantities
cc-model bare value quantities
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Microscopic coupled-channel model
where vertex interaction ?V is given in a
microscopic theory as
and integral LS type equation is given as
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if the interaction term is given as
the resonant interaction term can be
schematically written as
Then the resonant T-matrix is separated as
and the dressed propagator is
with
So the equation for the dressed vertex reads
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So, in such a type of a model we have two type of
quantities bare
and dressed
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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Checked for ? ? ?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 idea to identify bare quantities in a cc
  model with quark model ones is introduced
• The idea has not been proven.
• 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 results for a dynamic coupled-channel model
  are given yet

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I propose to make a comparison of N resonance
spectra with quark-model constituent model
predictions for a simpler case 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 very microscopic description of interaction
term
we solve the equivalent Dyson-Schwinger equation
for the Green function
with only effectively representing the
interaction term
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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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We require that the channel propagator is analytic
where qa(s) is the meson-nucleon cms momentum
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What should be identified with what?
Accepting the idea from photo-production we
identify bare quantities
with quark-model resonant states
and dressed
quantities with measurable quantities.
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Now, in case of N resonance parameters we have
to discuss two quantities
resonance mass and resonance width
We propose
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?

• Being aware that
• the interaction is not calculated but represented
  only effectively
• the available data base is far from sufficient

We expect to obtain only qualitative agreement.
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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
• Compared to
• Capstick-Roberts constituent quark model
  states

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Main 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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Comparison between
bare propagator pole s0 quark
model resonant states
and
dressed propagator poles
PDG parameters 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
Summary no visible problems details have to be
worked out

• All three lowest bare propagator poles can be
  identified with lowest QMRS (for the final
  solution there is a certain shift, but having in
  mind the quality of input ... no wonder)
• The obained dressed propagator poles only
  reasonably well correspond to experimental values
• The position of the third S11 pole is uncertain

MAIN PROBLEM insufficiency of
partial wave data SOLUTION
more inelastic channels have to be measured
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D13
pN elastic pN ? ?N
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D13
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D13
Summary no visible problems details have to be
worked out

• All three lowest bare propagator poles can be
  identified with lowest QMRS (for the final
  solution there is a certain shift, but having in
  mind the quality of input ... no wonder)
• The obained dressed propagator poles only
  reasonably well correspond to experimental
  values, because
• Either all three D13 poles are shifted
  downwards, or we have yet to detect another
  dressed scattering matrix pole at 1350 MeV

MAIN PROBLEM insufficiency of
partial wave data SOLUTION
more inelastic channels have to be measured
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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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P13

• Summary
• The quality of the fit is good
• Only one out of five predicted QMRS is needed
• The second bare propagator pole should be either
  identified with one of higher lying QMRS or will
  be shifted downwards when the data input is
  expanded with partial wave data from other
  channels

MAIN PROBLEM insufficiency of
partial wave data SOLUTION
more inelastic channels have to be measured
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P11
NOTORIOUSLY PROBLEMATIC ONE
pN elastic pN ? ?N
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P11
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P11

• Summary
• The quality of the fit is good
• Nucleon pole is introduced, results are
  insensitive to its presence
• one of experimentally confirmed SMPs, namely the
  N(1440) P11 state - Roper resonance, is not
  produced by any nearby bare propagator pole as it
  was the case for all other scattering-matrix
  poles it is generated differently.
• no bare propagator pole which would correspond to
  the 1.540 quark-model state is needed

MAIN PROBLEM insufficiency of
partial wave data SOLUTION
more inelastic channels have to be measured
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• Conclusions
• In coupled-channel models we have two type
  singularities at our disposal to be discussed
• bare scattering matrix poles and dressed
  scattering matrix poles
• The bare propagator poles in a CMB type
  coupled-channel model are consistent with
  constituent quark model resonant states
• 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
• Dressed scattering matrix poles are another story
  (extraction of scattering matrix pole, strong
  dissipation of Breit-Wigner parameters as
  reported in PDG...)

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