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A. Švarc
Rudjer Bošković Institute, Zagreb, Croatia
INT-09-3
The Jefferson Laboratory Upgrade to 12 GeV
(Friday, November 13, 2009)
Continuum ambiguities as a limitation factor in
single-channel PW analysis
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Continuum ambiguity is an old problem
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Tallahassee 2005
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1984
1978
1981
1978
Zgb
1985
Zgb
Zgb
1973
Today
Nothing much changed
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However, people are encountering problems when performing single channel PWA.
Illustration:
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Possible explanation of the problems: continuum ambiguities
because they have single channel fit
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However, people are (in principle) aware of the existence of continuum ambiguities!
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Pg. 5
Pg. 6
- Hoehler
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What does it mean “continuum ambiguity”?
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Simplified definition:
In a single-channel case, phase shifts (partial wave poles) are not always uniquely defined!
Unfortunately it turns out that this is the case as soon as inelastic channels open.
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Differential cross section (or any bilinear of scattering functions)
is not sufficient to determine the scattering amplitude:
if
then
The new function gives EXACTLY THE SAME CROSS
SECTION
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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
indeed lie upon or inside its unitary circle
HOW?
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These family of functions, though containing a continuum infinity of points,
are limited in extend.
The ISLANDS OF AMBIGUITY are created.
there exists a whole family of functions F , of limited magnitude but of infinite variety of functional form, which will
indeed lie upon or inside its unitary circle
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I M P O R T A N T
DISTINCTION
theoretical islands of ambiguity / experimental uncertainties
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Let us illustrate this on a simple example!
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The treatment of continuum ambiguity problems
1. How to obtain continuity in energy?2. How to achieve uniqueness?
The issues are:
In original publications several methods are suggested.
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However, there is another way to restore uniqueness:
by restoring unitarity in a coupled channel formalism
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Let us formulate what the continuum ambiguity problem means in the language of coupled channel
formalism
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Continuum ambiguity / T-matrix poles
Each analytic function is uniquely defined with its poles and cuts.
If an analytic function contains a continuum ambiguity it is not uniquely defined.
T matrix is an analytic function in s,t.
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 and importance 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.
Statement:
We need ALL channels, elastic AND as much inelastic ones as possible in order to uniquely define ALL scattering matrix poles.
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What is the procedure?
1. Having a coupled-channel formalism and fitting data only in one channel we may “mimic” single channel case.
2. By fitting one channel only we shall reveal those poles (resonant states) which dominantly couple to this channel.
3. Poles (resonant states) which do not couple to this channel will remain undetected.
4. Consequently, we have not been able to discover ALL analytic function poles, consequently the partial wave analytic function is ambiguous.
5. If we add data for the second inelastic channel, we constrain other set of poles which dominantly couple to this channel. This set of poles is overlapping with the first one, but not necessarily identical.
6. We have established a new, enlarged set of poles which is somewhat more constraining the unknown analytic function
7. We add new inelastic channels until we have found all scattering matrix poles, and uniquely identified the type of analytic PW function
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Example 1:
The role of inelastic channels in N (1710) P11
Published:
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All coupled channel models are based on solving Dyson-Schwinger integral type equations, and they all have the same general structure:
full = bare + bare * interaction* full
0 0G G G G
CMB coupled-channel model
0 0 0 0 0G G G G G G G
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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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Assumption: The imaginary part of the channel propagator is defined as:
2 2( ( ) )( ( ) )( )
4a
s M m s M mq s
s
where qa(s) is the meson-nucleon cms momentum:
And we require its analyticity through the dispersion relation:
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3434
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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We use:
1. CMB model for 3 channels: p N, h N, and dummy channel p2N 2. p N elastic T matrices , PDG: SES Ar06 3. p N ¨h N T matrices, PDG: Batinic 95
We fit:
1. πN elastic only2. p N ¨h N only3. both channels
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Results for extracted pole positions:
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Conclusions
1. Continuum ambiguities appear in single channel PWA, and have to be eliminated.
2. A new way, based on reinstalling unitarity is possible within the framework of couple-channel models.
3. T matrix poles, invisible when only elastic channel is analyzed, may spontaneously appear when inelastic channels are added.
4. 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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A few transparencies from NSTAR2005 talk:
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