A. ŠvarcA. Švarc

Rudjer Bošković Institute, Zagreb, CroatiaRudjer Bošković Institute, Zagreb, Croatia

PolesPoles as a link between as a link between QCDQCD and and scattering theoryscattering theory

(old and contemporary knowledge)(old and contemporary knowledge)

Several times here in GWU a statement has been made:Several times here in GWU a statement has been made:

It is clear that the poles are the only resonance signalIt is clear that the poles are the only resonance signal

However, it is still not clear whether:However, it is still not clear whether:

1.1.Is it really completely true?Is it really completely true?2.2.Is it a general knowledge?Is it a general knowledge?3.3.How are theHow are the poles poles to be extracted from experiment? to be extracted from experiment?4.4.IIs that statement adequately represented in the secondary s that statement adequately represented in the secondary literature? (meaning PDG)literature? (meaning PDG)

The main intention of this talk is to analyze the crucial ideas: The main intention of this talk is to analyze the crucial ideas: definition of resonances, poles and Breit-Wigner parameters.definition of resonances, poles and Breit-Wigner parameters.

I will give information about issues as I see them, and at I will give information about issues as I see them, and at least one important reference which can serve as a least one important reference which can serve as a

starting point for further reading. starting point for further reading.

The talk is going to be The talk is going to be referralreferral and not and not educational. educational.

The emphasis is going to be on existing, but sometimes The emphasis is going to be on existing, but sometimes forgotten knowledge about the field.forgotten knowledge about the field.

I What is a resonanceI What is a resonance

How do How do I see I see what is our main task?what is our main task?

ExperimentExperimentTheoryTheory

?

Matching pointMatching point

bound state

s

bound state

s

resonances

resonances

bound states

bound states

resonances

resonances

QCDQCD

What is a resonance in What is a resonance in QCDQCD??

What is a resonance in What is a resonance in experiment experiment??

What is a resonance in What is a resonance in experiment experiment??

From the intuitive (heuristic) definition to the mathematical formulationFrom the intuitive (heuristic) definition to the mathematical formulation

Heuristic definitionHeuristic definition

a.a. particle “gets trapped” (the “black hole” phenomenon)particle “gets trapped” (the “black hole” phenomenon)b.b. a direct scattering eventa direct scattering eventc.c. the lifetime of the particle-target system in the region of the lifetime of the particle-target system in the region of

interaction interaction is larger is larger than than the collision time the collision time in a direct in a direct collision processcollision process

This is a very heuristic definition.This is a very heuristic definition.

Question: Question: What is the What is the mathematical formulation mathematical formulation of of resonance signal?resonance signal? Where do the Breit-Wigner parameters come in?Where do the Breit-Wigner parameters come in?

What is a resonance signal?What is a resonance signal?

There is an uncertainty in literature.There is an uncertainty in literature.

A careful reader will notice that it is never explicitly stated: A careful reader will notice that it is never explicitly stated: the resonance is .......the resonance is .......

Instead, introducing and defining the resonance is always Instead, introducing and defining the resonance is always much more delicate.... much more delicate....

ExamplesExamples

Introducing Introducing (and NOT defining) (and NOT defining) resonances varies resonances varies from using from using rather rather undefined terms like:undefined terms like:

1.1. follow fromfollow from2.2. are associated withare associated with3.3. it is well knownit is well known 11.

22.

33.

Over strong statements that resonances are just a matter Over strong statements that resonances are just a matter of convention of convention “ ... simply an ad hoc hypothesis...” “ ... simply an ad hoc hypothesis...” ::

To full mathematical rigorTo full mathematical rigor

However, for me, the most transparent discussion is given in:However, for me, the most transparent discussion is given in:

"Adventures in Mathematical Physics" (Proceedings, Cergy-Pontoise "Adventures in Mathematical Physics" (Proceedings, Cergy-Pontoise 2006), Contemporary Mathematics, 447 (2007) 73-812006), Contemporary Mathematics, 447 (2007) 73-81

Both definitions of resonances are being used in the literature Both definitions of resonances are being used in the literature without full awareness that they are different, and that both are in without full awareness that they are different, and that both are in principle allowed. principle allowed.

This is a reason for numerous disputes and controversies.This is a reason for numerous disputes and controversies.

Knowing that we are dealing with the two equivalent Knowing that we are dealing with the two equivalent quantifications of the same phenomenon solves the issue. quantifications of the same phenomenon solves the issue.

The final answer to which of the two resonance definitions The final answer to which of the two resonance definitions should be used comes from analyzing what is precisely should be used comes from analyzing what is precisely calculated in QCD! calculated in QCD!

Is it a time delay or a resolvent poleIs it a time delay or a resolvent pole??

Let us remember: Let us remember: our task is not only describe resonances, buour task is not only describe resonances, butt describe them in a describe them in a way which is identical to QCD.way which is identical to QCD.

Scattering resonances:Scattering resonances:

They are aware that it is nowadays usual to identify resonances as They are aware that it is nowadays usual to identify resonances as resolvent resonancesresolvent resonances::

However they still advocate the idea of using scattering However they still advocate the idea of using scattering resonances insteadresonances instead:

Resolvent resonances:Resolvent resonances:

Concept of resolvent resonances is at length discussed inConcept of resolvent resonances is at length discussed in

Highlights:Highlights:

1.1. Equivalence of Hamiltonian eigenstates and resolvent polesEquivalence of Hamiltonian eigenstates and resolvent poles

2.2. The conservation of probability requires that the The conservation of probability requires that the resolvent poles lie in the complex energy plane resolvent poles lie in the complex energy plane

3.3. Importance of backgroundImportance of background

So, his final recommendation goes with Hamiltonian eigenstates – poles So, his final recommendation goes with Hamiltonian eigenstates – poles (resolvent resonance definition). (resolvent resonance definition).

What is a resonance in What is a resonance in QCDQCD??

Talk presented at the Workshop Talk presented at the Workshop "Light-cone Physics: Particles and Strings" at ECT* in Trento, Sep 3-11, "Light-cone Physics: Particles and Strings" at ECT* in Trento, Sep 3-11, 20012001

...solving the bound-state problem in gauge filed theory, particularly QCD......solving the bound-state problem in gauge filed theory, particularly QCD...

How is it done?How is it done?

Light-cone approachLight-cone approach

resonances resonances Hamiltonian proper valuesHamiltonian proper values

discrete light-cone quantizationdiscrete light-cone quantization

In all of these cases:In all of these cases:

Resonance Resonance areare identified with the Hamiltonian proper value identified with the Hamiltonian proper valuess

QCD is analyzing QCD is analyzing resolvent resonances resolvent resonances only!only!

Remember Exner and Lipovsky definition of resolvent resonance:Remember Exner and Lipovsky definition of resolvent resonance:

So the answer to our question is: So the answer to our question is:

POLES POLES

II How to extract T-matrix polesII How to extract T-matrix poles

Yesterday, Kanzu Nakayama discussed sets of poles Yesterday, Kanzu Nakayama discussed sets of poles coming from various analyses.coming from various analyses.

Let me show more precisely how I see the problem. Let me show more precisely how I see the problem.

Model 3Model 3

Model 2Model 2

Exp

erim

enta

l dat

a

Exp

erim

enta

l dat

a ba

se b

ase

Poles 1Poles 1

Poles 2Poles 2

Poles 3Poles 3

Under which conditions we have:Under which conditions we have:

Poles 1 = Poles 2 = Poles 3 ?Poles 1 = Poles 2 = Poles 3 ?

Is it enough that all models fit Is it enough that all models fit identicalidentical data base data base equallyequally well? well?

NO ! NO !

Illustration:Illustration:

Red line: Red line: Juelich modelJuelich model

Black line: Black line: Zagreb CMB fit to Juelich modelZagreb CMB fit to Juelich model

Red line:Red line:Complex Complex N branch pointN branch point no P11(1710)no P11(1710)

Black line:Black line:Real Real 22N branch point N branch point P11(1710) exists P11(1710) exists

In order to offer some insight into the problem we have used Zagreb In order to offer some insight into the problem we have used Zagreb CMB model to analyze a CMB model to analyze a collection collection of well known PW amplitudes of well known PW amplitudes UNDERSTANDING UNDERSTANDING

THEM ASTHEM AS PARTIAL WAPARTIAL WAVVE DATAE DATA

In that way we have effectively merged into one joint errorIn that way we have effectively merged into one joint error

1.1.error of fitting different input data bases anderror of fitting different input data bases and2.2. error due to different analytic structure of various models.error due to different analytic structure of various models.

We have to keep in mind that for all analyzed PWA input data We have to keep in mind that for all analyzed PWA input data base was almost identical, and the reduced base was almost identical, and the reduced 2 2 comparable, comparable, the major part of the obtained difference in pole positions is the major part of the obtained difference in pole positions is due to the different analytic structure of the particular model. due to the different analytic structure of the particular model.

We We did not expect to did not expect to reproduce the pole positions of each reproduce the pole positions of each individual analysis because our analytic structure is different individual analysis because our analytic structure is different from their, we just wanted to extract poles using on and the from their, we just wanted to extract poles using on and the same tool in each case.same tool in each case.

And have produced a sequel of preprints illustrating how And have produced a sequel of preprints illustrating how the method works for the S11 partial wave :the method works for the S11 partial wave :

1.1.

2.2.

Both publication are submitted to PRC.Both publication are submitted to PRC.

III Is the problem well presented in PDGIII Is the problem well presented in PDG

I personally believe that PDG is the most important secondary I personally believe that PDG is the most important secondary publication in physics publication in physics

because it is read by millions!because it is read by millions!

It is read by:It is read by:• professionals on a daily basis (just to see what is new professionals on a daily basis (just to see what is new

because they know it by heart)because they know it by heart)• experts in physics for getting reliable info about facts in experts in physics for getting reliable info about facts in

new fields they might be enteringnew fields they might be entering• professors in education to get the most precise, reliable, professors in education to get the most precise, reliable,

and worldly acceptable interpretation oand worldly acceptable interpretation off some possibly some possibly controversial statements found in literaturecontroversial statements found in literature

• students to get the condensed, precise new knowledge of students to get the condensed, precise new knowledge of unknown phenomenaunknown phenomena

• layman to see what is the most current value of certain layman to see what is the most current value of certain constant, and how the experts in physics thinkconstant, and how the experts in physics think

And I believe that the responsibility of PDG is to sytisfy them all.And I believe that the responsibility of PDG is to sytisfy them all.

Therefore it is its duty to beTherefore it is its duty to be• PRECISE PRECISE

(define all phenomena, all procedures, all assumptions…)(define all phenomena, all procedures, all assumptions…)• CONDENSEDCONDENSED• CRITICALCRITICAL• EXACTEXACT

Is it really so today in baryon spectroscopy sector?Is it really so today in baryon spectroscopy sector?

I am afraid IT IS NOT.I am afraid IT IS NOT.

The reasons for such a statement of mine is that in spite of strong The reasons for such a statement of mine is that in spite of strong recent progress in the field this section recent progress in the field this section has barely changed in last 20 has barely changed in last 20 years: years:

• the definition of resonance the definition of resonance is NEVER GIVEN !is NEVER GIVEN ! PDG is PDG is presenting chapters and chapters of the book relying on presenting chapters and chapters of the book relying on heuristic understanding of a reader what a resonance really is, heuristic understanding of a reader what a resonance really is, and this creates ambiguities. and this creates ambiguities.

• the difference between poles and Breit-Wigner parameters is the difference between poles and Breit-Wigner parameters is never explicitly givennever explicitly given

• ththe definition, importance and role of poles is never givene definition, importance and role of poles is never given• •

Examples:Examples:

1.1. I am for quite some time objecting to the way how the nucleon I am for quite some time objecting to the way how the nucleon resonances chapter is being written with respect to defining the resonances chapter is being written with respect to defining the resonance parameters (mass and width)resonance parameters (mass and width)

TThe situation is quite clear in the meson part:he situation is quite clear in the meson part:

However, it is NOT for the nucleon sector.However, it is NOT for the nucleon sector.

And what is interesting to me is that one of the leading PDG And what is interesting to me is that one of the leading PDG experts for nucleon resonances is since 1999 aware of the problem experts for nucleon resonances is since 1999 aware of the problem

Ron has announced some changes….Ron has announced some changes….

Breit-Wigner parametersBreit-Wigner parameters Pole positionsPole positionsPDG20PDG201010

The situation remains the same even today!The situation remains the same even today!

III: How to proceed?III: How to proceed?

In Trento 2011In Trento 2011

I have also discussed this issue, and I have shown the following I have also discussed this issue, and I have shown the following two slides: two slides:

And I am happy that we have a chance to do it on Thursday!And I am happy that we have a chance to do it on Thursday!