The importance of inelastic channels in eliminating continuum ambiguities in pionnucleon partial wav

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The importance of inelastic channels in
eliminating continuum ambiguities in pion-nucleon
partial wave analyses

• Alfred varc
• Ruder Bokovic Institute
• Croatia

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constraints from fixed t-analyticity resolve
the ambiguities
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The effects of discreet and continuum ambiguities
were separated .
Dispersion relations were used to solve
ambiguities and to derive
constraints
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2005
1973
1975
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1973
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1985.
2005
1979
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1976
2005
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What does it mean continuum ambiguity?
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Differential cross section is not sufficient to
determine the scattering amplitude
if
The new function gives EXACTLY THE SAME CROSS
SECTION
then
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S matrix unitarity .. conservation of flux

RESTRICTS THE PHASE
elastic region . unitarity relates real and
imaginary part of each partial
wave equality constraint
each partial wave must lie upon its unitary circle
inelastic region . unitarity provides only an
inequality
constraint between real and imaginary part
each partial wave must lie upon or inside its
unitary circle
there exists a whole family of functions F , of
limited magnitude but of infinite variety of
functional form, which will behave exactly like
that
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there exists a whole family of functions F , of
limited magnitude but of infinite variety of
functional form, which will behave exactly like
that
These family of functions, though containing a
continuum infinity of points, are limited in
extend. The ISLANDS OF AMBIGUITY are created.
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I M P O R T A N T
Once the three body channels open up, this way of
eliminating continuum ambiguities (elastic
channel arguments) become in principle impossible
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I M P O R T A N T
DISTINCTION theoretical islands of
ambiguity / experimental uncertainties
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The treatment of continuum ambiguity problems

• Constraining the functional form F -
  mathematical problem
• Implementing the partial wave T matrix
  continuity
• (energy smoothing and search for uniqueness)

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a.
Finding the true boundaries of the phase function
F (z) is a very difficult problem on which the
very little progress has been made.
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b.
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Let us formulate what the continuum ambiguity
problem is in the language of
coupled channel formalism
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Continuum ambiguity / T-matrix poles
T matrix is an analytic function in s,t.
Each analytic function is uniquely defined with
its poles and cuts.
If an analytic function contains a continuum
ambiguity it is not uniquely defined.
If an analytic function is not uniquely defined,
we do not have a complete knowledge about its
poles and cuts.
Consequently
fully
constraining poles and cuts means eliminating
continuum ambiguity
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Basic idea
we want to demonstrate the role of
inelastic channels in fully constraining the
poles of the partial wave T-matrix,
or, alternatively said, for
eliminating continuum ambiguity which arises if
only elastic channels a considered.
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We want as well show that supplying only scarce
information for EACH channel is MUCH MORE
CONSTRAINING then supplying
the perfect information in ONE channel.
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Coupled channel T matrix formalism

• unitary
• fully analytic

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T-matrix poles are connected to the bare
propagator poles, but shifted with the self
energy term !
Important
real and imaginary parts of the self
energy term are linked because of analyticity
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• Constraining data
• Elastic channel
• Pion elastic VPI SES solution FA02
  Karlsruhe - Helsinki
  KH 80

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• Inelastic channel
• recent
• pN h N CC PWA Pittsburgh/ANL 2000
• however NO P11 is offered !
• older
• p N hN CC PWA Zagreb/ANL
  95/98
• p N p2 N CC PWA Zagreb/ANL
  95/98
• gives P11

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• Both three channel version of CMB model

• p N elastic T matrices p N h N data dummy
  channel

• Pittsburgh VPI data dummy channel
• Zagreb KH80 data dummy channel

• fitted all partal waves up to L 4

• Pittsburgh offers S11 only
• Zagreb offers P11 as well
  (nucl-th/9703023)

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S11
Let us compare Pittsburgh/ Zagreb S11
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Pittsburgh S11 is taken as an experimentally
constrained partial wave
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So we offer Zagreb P11 as the experimentally
constrained partial wave as well.
P11
from nucl-th/9703023
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Two-channel-model
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STEP 1

• Number of channels
  2
• (pion elastic effective)
• Number of GF propagator poles
  3
• (2 background poles 1 physical pole)
• ONLY ELASTIC CHANNEL IS FITTED

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• elastic channels is reproduced perfectly
• inelastic channel is reproduced poorly
• we identify one pole in the physical region

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STEP 2
Number of channels
2 (pion elastic
effective) Number of GF propagator poles
3 (2 background poles
1 physical pole) ONLY INELASTIC CHANNEL IS
FITTED
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• inelastic channel is reproduced perfectly
• elastic channel is reproduced poorly
• we identify two poles in the physical region,
  Roper and 1700 MeV

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STEP 3
Number of channels
2 (pion elastic
effective) Number of GF propagator poles
3 (2 background poles
1 physical pole) ELASTIC INELASTIC CHANNEL ARE
FITTED
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• elastic channel is reproduced OK
• inelastic channel is reproduced tolerably
• we identify two poles in the physical region,
  but both are in the Roper
  resonance region

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We can not find a single pole solution which
would simultaneously reproduce ELASTIC AND
INELASTIC CHANNELS
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STEP 4
Number of channels
2 (pion elastic
effective) Number of GF propagator poles
4 (2 background poles
2 physical poles) ONLY ELASTIC CHANNEL IS FITTED
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• elastic channels is reproduced perfectly
• inelastic channel is reproduced poorly
• we identify two poles in the physical region,
  Roper one above 2200 MeV

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STEP 5
Number of channels
2 (pion elastic
effective) Number of GF propagator poles
4 (2 background poles
2 physical poles) ONLY INELASTIC CHANNEL IS
FITTED
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We have found two possible solutions which
differ significantly in channels which are not
fitted
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Solution 1
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Solution 2
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• For both solutions
• elastic channel is poorly reproduced, AND
  differs for both solutions
• inelastic channel is OK
• For both solutions we identify two poles in the
  physical region, Roper 1700 MeV

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STEP 6
Number of channels
2 (pion elastic
effective) Number of GF propagator poles
4 (2 background poles
2 physical poles) ELASTIC INELASTIC CHANNELS
ARE FITTED
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We offer two possible solutions which differ
significantly in channels which are not fitted
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Solution 1
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Solution 2
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• For both solutions
• elastic channel is acceptably reproduced, AND
  differs for both solutions
  BUT ADDITIONAL STRUCTURE IN
  THE ELASTIC CHANNEL APPEARES IN THE ENERGY RANGE
  OF1700 MeV
• inelastic channel is OK
• For both solutions we identify two poles in the
  physical region, Roper 1700 MeV

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• An observation
• The structure in elastic channel, required by
  the presence of inelastic channels, appears
• exactly where error bars of the Fa02 solution are
  big
• exactly in the place where KH80 shows a structure
  not observed in the FA02

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Three-channel-model
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THE SAME STORRY AS FOR TWO CHANNELS BUT MUCH
MORE FINE TUNING IS NEEDED (better input us
required)
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STEP 7
Number of channels
3 (pion elastic
effective) Number of GF propagator poles
4 (2 background poles
2 physical poles) ONLY ELASTIC CHANNELS IS
FITTED
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STEP 8
Number of channels
3 (pion elastic
effective) Number of GF propagator poles
4 (2 background poles
2 physical poles) ELASTIC INELASTIC CHANNELS
ARE FITTED
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We offer three solutions
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Solution 1
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Solution 2
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Solution 3
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Conclusions

• T matrix poles, invisible when only elastic
  channel is analyzed, spontaneously appear in the
  coupled channel formalism when inelastic channels
  are added.
• It is demonstrated that
• the N(1710) P11 state exists
• the pole is hidden in the continuum ambiguity of
  VPI/GWU FA02
• it spontaneously appears when inelastic channels
  are introduced in addition to the elastic ones.

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How do we proceed?

• Instead of using raw data we have decided to
  
  represent them in a form of partial wave
  T-matrices (single channel PWA,
  something else
• We use them as a further constraint in a CC_PWA

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I
II
III
A call for help Anyone who has some kind of
partial wave T-matrices, regardless of the way
how they were created please sent it to us,
so that we could,

within the framework of our formalism,
establish which poles are responsible for
their shape.
Partial wave T- matrices
Experiment
Different data sets
BRAG ?
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Topics to be resolved

• The background should be introduced in a
  different way, because the recipe of simulating
  the background contribution with two distant
  poles raises severe technical problems in fitting
  procedure.
• The formalism should be re-organized in such a
  way that the T matrix poles, and not a bare
  propagator poles become a fitting parameter.
• The existence of other, low star PDG resonances,
  should be checked.