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

Continuum ambiguity is an old problem

Tallahassee 2005

1984

1978

1981

1978

Zgb

1985

Zgb

Zgb

1973

Today

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!

Pg. 5

Pg. 6

- Hoehler

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

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?

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

I M P O R T A N T

DISTINCTION

theoretical islands of ambiguity / experimental uncertainties

Let us illustrate this on a simple example!

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.

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

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

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. 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

Example 1:

The role of inelastic channels in N (1710) P11

Published:

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

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-resonance mixing matrix

bare particle propagatorchannel propagator

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:

3434

0 0G G G G

we obtain the full propagator G by solving Dyson-Schwinger equation

where

we obtain the final expression

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

Results for extracted pole positions:

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.

A few transparencies from NSTAR2005 talk: