Transcript "Continuum ambiguities as a limitation factor in single-channel PW analysis"

Continuum ambiguities as a
limitation factor in
single-channel PW analysis
A. Švarc
Rudjer Bošković Institute, Zagreb, Croatia
INT-09-3
The Jefferson Laboratory Upgrade to 12 GeV
(Friday, November 13, 2009)
Continuum ambiguity is an old problem
Tallahassee 2005
Today
1984
1978
1985
1973
1981
Zgb
1978
Zgb
Zgb
Nothing much changed
However, people are encountering problems when
performing single channel PWA.
Illustration:
Possible explanation of the problems:
continuum ambiguities
because they have single channel fit
However, people are (in principle) aware of the existence of
continuum ambiguities!
- Hoehler
Pg. 5
Pg. 6
What does it mean “continuum ambiguity”?
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.
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
S – matrix unitarity …………….. conservation of flux
RESTRICTS THE PHASE
HOW?
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
upon or inside its unitary circle
upon or inside its unitary circle
These family of functions, though containing a continuum infinity
of points,
are limited in extend.
The ISLANDS OF AMBIGUITY are created.
I M P O RTA N T
DISTINCTION
theoretical islands of ambiguity / experimental uncertainties
Let us illustrate this on a simple example!
The treatment of continuum ambiguity
problems
The issues are:
1. How to obtain continuity in energy?
2. How to achieve uniqueness?
In original publications several methods
are suggested.
However, there is another way to
restore uniqueness:
by restoring unitarity in a coupled
channel formalism
Let us formulate what the continuum ambiguity
problem means in the language of coupled channel
formalism
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
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.
What is the procedure?
1.
2.
3.
4.
5.
6.
7.
Having a coupled-channel formalism and fitting data only in
one channel we may “mimic” single channel case.
By fitting one channel only we shall reveal those poles
(resonant states) which dominantly couple to this channel.
Poles (resonant states) which do not couple to this channel
will remain undetected.
Consequently, we have not been able to discover ALL
analytic function poles, consequently the partial wave
analytic function is ambiguous.
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.
We have established a new, enlarged set of poles which is
somewhat more constraining the unknown analytic function
We add new inelastic channels until we have found all
scattering matrix poles, and uniquely identified the type of
analytic PW function
Example 1:
The role of inelastic channels in N (1710) P11
Published:
CMB coupled-channel model
All coupled channel models are based on solving DysonSchwinger integral type equations, and they all have the same
general structure:
full = bare + bare * interaction* full
G  G0  G0   G
G  G0  G0   G0  G0   G0   G
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.
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
Assumption: The imaginary part of the channel propagator is defined as:
And we require its analyticity through the dispersion
relation:
where qa(s) is the meson-nucleon cms momentum:
( s  ( M  m)2 )( s  ( M  m) 2 )
qa ( s) 
4s
we obtain the full propagator G by solving Dyson-Schwinger equation
G  G0  G0   G
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:
πN elastic only
2. p N h N only
3. both channels
1.
Results for extracted pole positions:
Conclusions
1.
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4.
Continuum ambiguities appear in single channel PWA, and have to
be eliminated.
A new way, based on reinstalling unitarity is possible within the
framework of couple-channel models.
T matrix poles, invisible when only elastic channel is analyzed, may
spontaneously appear 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.
A few transparencies from NSTAR2005 talk: