Bare propagator poles in coupled-channel
models
(Possible link between microscopic theories and
phenomenological models)
Alfred Švarc
Ruđer Bošković Institute
Croatia
1
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 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
2
The question are:
1. Can we formulate the problem here exactly?
2. Can we make a step forwards towards giving a
competent answer to the existing controversy?
3
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 offshell effects are unmeasurable because of first field-theoretical principles
4
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
5
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
6
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?
7
The formulation of hadron spectroscopy program
Höhler – Landolt Bernstein
A. Švarc, 2ndPWA Workshop, Zagreb 2005
8
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.
9
Up to now:
10
However, coupled channel models, based on solving Dyson-Schwinger
integral type equations having the general structure
full = bare + bare * interaction* full
G  G0  G0   G
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?
11
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?
12
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):
cc-model bare value
quantities

quark-model
quantities
Question: Can the idea be justified?
Details given in
13
The idea has been repeated since:
14
15
2004
16
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.
17
I will now give a short preview of the essential from:
18
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)
19
UP TO NOW
quark model
resonant states

scattering
matrix poles
Problems for transition
amplitudes
Proposed way out
20
Applied to Δ → γN helicity amplitudes
21
Extension to the full N* resonance spectra is proposed in
Matsuyama, Sato and Lee, Physics Reports 439 (2007):
However it is not yet done:
22
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
23
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.
24
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.
25
We represent the full T-matrix in the form where the channel-resonance
interaction is not calculated but effectively parameterized:
channel propagator

channel-resonance
mixing matrix

bare particle
propagator
26
we obtain the full propagator G by solving Dyson-Schwinger equation
G  G0  G0   G
where
we obtain the final expression
27
What should be identified with what?
28
Following the idea from photoproduction:
bare propagator pole position
imaginary part of the dressed
propagator pole


mass of a quark-model resonant state
decay width
29
What is our aim?
To establish if there is any regular pattern of behavior .
30
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
32
The intention is to ask for the absolute minimum!
To see if the interpretation of bare propagator poles as quarkmodel 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
33
The comparison is done for lowest partial waves
S11 , P11 , P13 and D13
34
Let us show the two lowest parity odd states:
S11
 2
1

and D13
 2
3

35
S11
 2
1

 R2  3.4
 R2  2.6
36
S11
 2
1

dressed pole
PDG
quark model
resonant state
constrained fit bare
propagator mass
free fit bare
propagator mass
37
S11
 2
1

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
38
D13
 2
3

πN elastic
πN → ηN
 R2  2.0
 R2  1.5
39
D13
 2
3

40
1.590
1.753
1.972
2.162
41
Let us show the two lowest parity even states states:
P11
 2
1

and P13
 2
3

Problems appear
42
P13
 2
3

πN elastic
πN → ηN
43
P13
 2
3

44
1.725
1.922
2.220
45
P11
 2
1

NOTORIOUSLY PROBLEMATIC ONE
πN elastic
πN → ηN
46
P11
 2
1

47
1.612
1.728
2196
48
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
49
Final question to be answered here:
What is the correspondence between bare propagator
poles in general and hadron structure calculations?
50