A. Švarc Rudjer Bošković Institute, Zagreb, Croatia

A. ŠvarcA. ŠvarcRudjer 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)

HADRONIC RESONANCES, Bled (Slovenia), July 1-8, 2012

It It seems to beseems to be clear that the poles are the only resonance signal 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)


Therefore, in my talk I will address three topics:Therefore, in my talk I will address three topics:

I What is a resonanceI What is a resonance

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

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

The main intention of this talk is to analyze the crucial iThe main intention of this talk is to analyze the crucial issuesssues::

• DDefinition of resonances, poles and Breit-Wigner parametersefinition of resonances, poles and Breit-Wigner parameters• Importance of inelastic channelsImportance of inelastic channels• New experimentsNew experiments• Changes in PDGChanges in PDG

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 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 states

bound states

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

Extensive reviewExtensive review


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 (and NOT defining) resonances varies Introducing (and NOT defining) resonances varies from using rather undefined terms like:from using rather 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 sometimes sometimes without full awareness that they are different, and that without full awareness that they are different, and that both are in principle allowed. both are in 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.

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.


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??

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



Complex poles and non-hermitian operatorsComplex poles and non-hermitian operators



Complex poles and non-hermitian operatorsComplex poles and non-hermitian operators


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


Hamiltonian proper valuesHamiltonian proper values

Remember Remember Dalitz-MoorhouseDalitz-Moorhouse

polespoles

discrete light-cone quantizationdiscrete light-cone quantization


For the compilation of other approaches see ref.For the compilation of other approaches see ref.


At this Workshop we have seen that LQCD can also directly give At this Workshop we have seen that LQCD can also directly give phase shifts using Lüscher’s approach!phase shifts using Lüscher’s approach!


So, in principle:So, in principle:

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

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


II-b. Highlights from CamogliII-b. Highlights from Camogli

II -a Importance of inelastic channelsII -a Importance of inelastic channels

Model 3Model 3

Model 2Model 2

Expe

rimen

tal d

ata

Expe

rimen

tal d

ata b

ase

bas

e

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 ?



II -a Importance of inelastic channelsII -a Importance of inelastic channels



What is the procedure?What is the procedure?

1.1.Having a coupled-channel formalism and fitting data only in one Having a coupled-channel formalism and fitting data only in one channel will “mimic” single channel case. channel will “mimic” single channel case. 2.2.By fitting one channel only we shall reveal those poles (resonant By fitting one channel only we shall reveal those poles (resonant states) which dominantly couple to this channel.states) which dominantly couple to this channel.3.3.Poles (resonant states) which do not couple to this channel will Poles (resonant states) which do not couple to this channel will remain undetected.remain undetected.4.4.Consequently, we have not been able to discover ALL analytic Consequently, we have not been able to discover ALL analytic function poles, consequently the partial wave analytic function is function poles, consequently the partial wave analytic function is ambiguous.ambiguous.5.5.If we add data for the second inelastic channel, we constrain other If we add data for the second inelastic channel, we constrain other set of poles which dominantly couple to this channel. This set of set of poles which dominantly couple to this channel. This set of poles is overlapping with the first one, but not necessarily identical. poles is overlapping with the first one, but not necessarily identical. 6.6.We have established a new, enlarged set of poles which is We have established a new, enlarged set of poles which is somewhat more constraining the unknown analytic functionsomewhat more constraining the unknown analytic function7.7.We add new inelastic channels until we have found all scattering We add new inelastic channels until we have found all scattering matrix poles, and uniquely identified the type of analytic PW functionmatrix poles, and uniquely identified the type of analytic PW function

The Jefferson Laboratory Upgrade to 12GeV (INT 09-3), Seattle, 7-16 Nov.2009.


Example 1:Example 1:

The role of inelastic channels in The role of inelastic channels in N (1710) PN (1710) P11 11

Published: Published:



AllAll c coupled channel modelsoupled channel models are based on solving Dyson-Schwinger integral are based on solving Dyson-Schwinger integral type equations, and they all have the same general structure:type equations, and they all have the same general structure:

full = bare + bare * interaction* fullfull = bare + bare * interaction* full

0 0G G G G

CMB coupled-channel modelCMB coupled-channel model

0 0 0 0 0G G G G G G G


Carnagie-Melon-Berkely (CMB) modelCarnagie-Melon-Berkely (CMB) model

Instead of solving Lipmann-Schwinger equation of the type:Instead of solving Lipmann-Schwinger equation of the type:

with microscopic description of interaction termwith microscopic description of interaction term

we solve the equivalent Dyson-Schwinger equation for the Green functionwe solve the equivalent Dyson-Schwinger equation for the Green function

with representing the with representing the whole whole interaction term interaction term effectivelyeffectively..


We represent the full T-matrix in the form where the We represent the full T-matrix in the form where the channel-resonance channel-resonance interaction interaction is not calculated but effectively parameterized:is not calculated but effectively parameterized:

channel-resonance mixing matrix

bare particle propagatorchannel propagator

Assumption: Assumption: The imaginary part of the channel propagator is defined as:

2 2( ( ) )( ( ) )( )4a

s M m s M mq ss

where qa(s) is the meson-nucleon cms momentum:

And we require its analyticity through the dispersion relation:


505050

0 0G G G G

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

where

we obtain the final expression


We use:We use:

1.1.CMB model for 3 channels:CMB model for 3 channels: N, N, N, and dummy channel N, and dummy channel .. N elastic T matrices , N elastic T matrices , PDG: SES Ar06 PDG: SES Ar06 3.3. NN N T matrices, N T matrices, PDG:PDG: Batinic 95Batinic 95

We fit:We fit:

1.1.πN πN elastic onlyelastic only.. NN N only N only3.3.both channelsboth channels



Results for extracted pole positions:Results for extracted pole positions:

The Jefferson Laboratory Upgrade to 12GeV (INT 09-3),

Seattle, 7-16 Nov.2009.


ConclusionsConclusions

1.1. Continuum ambiguities appear in single channel PWA, and have to Continuum ambiguities appear in single channel PWA, and have to be eliminated.be eliminated.

2.2. A new way, based on reinstalling unitarity is possible within the A new way, based on reinstalling unitarity is possible within the framework of couple-channel models.framework of couple-channel models.

3.3. T matrix poles, invisible when only elastic channel is analyzed, may T matrix poles, invisible when only elastic channel is analyzed, may spontaneously appear when inelastic channels are added.spontaneously appear when inelastic channels are added.

4.4. It is demonstrated thatIt is demonstrated that:: the N(1710) Pthe N(1710) P11 11 state state existsexists thethe pole is hidden in the continuum ambiguity of pole is hidden in the continuum ambiguity of VPI/VPI/GWU FA02GWU FA02 it it spontaneously appears when inelastic channels are spontaneously appears when inelastic channels are

introduced in addition to the elastic ones. introduced in addition to the elastic ones.


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

Black line: Zagreb CMB fit to Juelich modelBlack line: Zagreb CMB fit to Juelich modelReal Real 22N branch point N branch point P11(1710) exists P11(1710) exists


However : PRELIMINARY – private information M. DöringHowever : PRELIMINARY – private information M. Döring

When including inelastic When including inelastic ππN → KN → KΛΛ in a CC fit in a CC fit P11(1710) is P11(1710) is reinstatedreinstated

Example Example 22::

We propose one way to extract poles from”experiment”: We propose one way to extract poles from”experiment”: to use to use Zagreb CMB model to analyze Zagreb CMB model to analyze „world” „world” collection of well known PW amplitudes collection of well known PW amplitudes

UNDERSTANDING THEM AS UNDERSTANDING THEM AS PARTIAL WAVE DATAPARTIAL WAVE 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.

6-th PWA Workshop, GWU, Washington DC, 22-30 June

2011.

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.


2011.


II-b. Highlights from CamogliII-b. Highlights from Camogli



On Breit-Wigner vs. pole parametersOn Breit-Wigner vs. pole parameters

Question: Question: Why using pole parameters when fitting with BW parameters Why using pole parameters when fitting with BW parameters give equally good give equally good χχ2 2 ??

Theory discussion session summary(mostly mesons)

J.R.PeláezU. Complutense, Madrid

On the use of BW and other stuff

• But my kids can also draw a perfect line interpolating all your data without learning anything about Physics.

• You can use whatever you want to fit your data….

• BUT: If you want to extract POLES or RESONANCE properties you need the correct analytic structure, unitarity, etc… otherwise it might be easily VERY WRONG

• And their χ2 , will be better than yours…!More seriously… see M. Williams talk

On the use of BW and other stuff

• REFRAIN from making Physical or POLE interpretations unless you are sure your parametrizations make sense in the complex plane (analyticity), are unitary, etc…

• Thus, if you insist in using whatever you want to fit your data….

• It might be useful to make it public and available for others because you want to parametrize the data somehow so that anybody can use it, compare to it, etc…

Fine… but.


Contribution for the white paper: ŠvarcContribution for the white paper: Švarc


On partial waves vs. full experimental data setOn partial waves vs. full experimental data set

Question: Question: Should one use experimental data base or partial waves in a Should one use experimental data base or partial waves in a form of partial wave data?form of partial wave data?


A few transparencies from NSTAR2005 talk:



On new experimentsOn new experiments


Why?Why? Inelastic channel experiments are old and contraversial!

ThereforeTherefore





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.6-th PWA Workshop, GWU, Washington DC, 22-30 June

2011.

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


2011.

At ATHOS 2012 it has been reported that signiificant changes have At ATHOS 2012 it has been reported that signiificant changes have been done!been done!

1.1. Prof. Eberhard Klempt replaced Prof. G. Hoehler as being Prof. Eberhard Klempt replaced Prof. G. Hoehler as being responsible person for hadronic resonance sectorresponsible person for hadronic resonance sector

Now we have Now we have R. Workman and E. KlemptR. Workman and E. Klempt

2.2. He has introduced significant changesHe has introduced significant changesi.i. Changed the introductionChanged the introductionii.ii. Added new resonancesAdded new resonancesiii.iii. Accepted some comments from other fieldsAccepted some comments from other fieldsiv.iv. Opened the possible communication link to PDGOpened the possible communication link to PDG






2011.

The f0(600) or “sigma” in PDG 1996-2010

M=400-1200 MeVΓ=500-1000 MeV

Is f0(500) or “sigma” in PDG 2012M=400-1200 MeVΓ=500-1000 MeV

M=400-550 MeVΓ=400-700 MeV

DRAMMATIC AND LONG AWAITED CHANGE ON “sigma” RESONANCE !!

And minor correction for the f0(980) mass & more conservative uncertainties

MeV20990MeV10980 MM


Documents

A. Švarc Rudjer Bošković Institute, Zagreb, Croatia