Poles of Poles of PWDPWDataata andand PW PWAAmplitudesmplitudes

in Zagreb modelin Zagreb model

A. ŠvarcA. Švarc, S. Ceci, B. Zauner, S. Ceci, B. Zauner

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

M. Hadžimehmedović, H. Osmanović, J. Stahov M. Hadžimehmedović, H. Osmanović, J. Stahov Univerzity of Tuzla, Tuzla Univerzity of Tuzla, Tuzla

Bosn Bosniaia and Herzegovina and Herzegovina

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

ExperimentExperiment QuarksQuarks

?

Matching pointMatching point

structu

res

structu

res

bound states

bound states

resonances

resonances

HHööhlehlerr – Landolt Bernstein 1984. – Landolt Bernstein 1984.

Burkert – Lee 200Burkert – Lee 2004.4.

Ceci, Svarc, Zauner 200Ceci, Svarc, Zauner 20055..

Svarc 2004 Svarc 2004

HHööhlehlerr – Landolt Bernstein 1984. – Landolt Bernstein 1984.

HHööhlehlerr – Landolt Bernstein 1984. – Landolt Bernstein 1984. explicit analytic form introducedexplicit analytic form introduced

o Phenomenological T-matrixPhenomenological T-matrix• CMU-LBLCMU-LBL• ZagrebZagreb• Argonne-PittsburghArgonne-Pittsburgh

o effective Lagrangianeffective Lagrangian• EBACEBAC• JuelichJuelich• Dubna-Mainz-Taipei (DMT)Dubna-Mainz-Taipei (DMT)• GiessenGiessen

o Chew-Mandelstam K-matrixChew-Mandelstam K-matrix• GWU/VPIGWU/VPI

PWDPWD

PWAPWA

Difference between Difference between PWDPWD and and PWAPWA

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

ExperimentExperiment QuarksQuarks

structu

res

structu

res

bound states

bound states

resonances

resonances

Breit-Wigner Breit-Wigner parametersparameters

Pole Pole parametersparameters

Phase shiftsPhase shifts

What is “better”:What is “better”:

• Breit-Wigner parametersBreit-Wigner parameters oror• Pole parametersPole parameters

The advantages and drawbacksThe advantages and drawbacks

Breit – Wigner parameters:Breit – Wigner parameters:

Advantages:Advantages:

• defined on the real axesdefined on the real axes• simple to calculatesimple to calculate

Harry Lee Harry Lee BRAG 2001BRAG 2001

Drawbacks:Drawbacks:

• dependence on the choice of field dependence on the choice of field variablesvariables

• model dependent model dependent (background definition)(background definition)

A simple illustration A simple illustration of BW model of BW model dependence:dependence:

Pole parametersPole parameters

AdvantagesAdvantages

• invariant with respect to the choice invariant with respect to the choice of field variablesof field variables

• model independentmodel independent• less model dependentless model dependent

DrawbacksDrawbacks

• hard to get because they lie in the complex energy planehard to get because they lie in the complex energy plane

Why?Why?

The dependence upon background parameterization is a The dependence upon background parameterization is a well know fact, well know fact, but nothing has been done in PDG yet. but nothing has been done in PDG yet.

PDG makes an average of all BW values, PDG makes an average of all BW values, regardless of regardless of the way background has been introduced. the way background has been introduced. This introduces an additional systematic error. This introduces an additional systematic error.

Extraction of Breit-Wigner parametersExtraction of Breit-Wigner parameters

Suggestion by Lothar Tiator:Suggestion by Lothar Tiator:

We all know that BW positions and parameters are We all know that BW positions and parameters are not well not well defined defined but many people believe that but many people believe that within some within some uncertaintyuncertainty they can be given, and are very useful. Therefore they can be given, and are very useful. Therefore I also tend to stick with them and would propose to keep I also tend to stick with them and would propose to keep them in PDG in the future. them in PDG in the future. But only if we can give some But only if we can give some proper definition and methods for extraction.proper definition and methods for extraction.

Suggestion by Suggestion by Harry LeeHarry Lee::

Each PWA has Each PWA has a specific assumptiona specific assumption on the analytic form. on the analytic form.

Idea:Idea:Let us use CMB formalism to analyze available PWD and PWALet us use CMB formalism to analyze available PWD and PWA using using one and the same one and the same analytic formanalytic form..

We propose:We propose:

1.1. To use To use new PWD or PWA new PWD or PWA in addition to the in addition to the existing existing ones ones and look for and look for the shift of poles the shift of poles (shift of present (shift of present ones, appearance of the new ones)ones, appearance of the new ones)

2.2. To To useuse ONE analytic ONE analytic form (Zagreb CMB) for form (Zagreb CMB) for ALL ALL EXISTING PWA and PWDEXISTING PWA and PWD, and:, and:a.a. extract poles from a particular PWD or PWA using extract poles from a particular PWD or PWA using

Zagreb CMB and compare the outcome with the Zagreb CMB and compare the outcome with the original resultoriginal result

b.b. compare agreement of poles of compare agreement of poles of ALL EXISTING PWA ALL EXISTING PWA and PWD and PWD in order in order to to avoid systematic error because avoid systematic error because of differences in analytic forms. of differences in analytic forms.

Extraction of pole parametersExtraction of pole parameters

WarningWarning::

When we analyze When we analyze particular particular PWA, we do not say that we shall PWA, we do not say that we shall exactly reproduce exactly reproduce pole positions given by thpole positions given by thatat particular model.particular model.

Results Results may differmay differ, and the difference will show the significance , and the difference will show the significance of analytic form of analytic form chosen to representchosen to represent on-energy shell da on-energy shell datta.a.

So, we are not checking if the pole positions in a certain model So, we are not checking if the pole positions in a certain model are correctly extracted, but rather seting up the way how to are correctly extracted, but rather seting up the way how to quantify the comparison of different curves.quantify the comparison of different curves.

1. 1. … to use … to use new PWD or PWA .... new PWD or PWA ....

Initial attempts ...Initial attempts ...

In details In details repeatedrepeated at NSTAR2005 - Tallahassee at NSTAR2005 - Tallahassee

Technical problems on Zagreb side......Technical problems on Zagreb side......

2.2. Using CMB to analyze a world collection of PWD and PWA Using CMB to analyze a world collection of PWD and PWA and eliminate model assumptions on analytic formand eliminate model assumptions on analytic form

• Formulated at BRAG2007Formulated at BRAG2007• Research in progressResearch in progress

REMARKREMARK

Technical problems in Zagreb code are now eliminated. Technical problems in Zagreb code are now eliminated. Code is running under LINUX. It is transferable from Code is running under LINUX. It is transferable from machine to machine, and is an open source code. machine to machine, and is an open source code.

Adjustments and improvements can be done. Adjustments and improvements can be done. (I can demonstrated how the code works during workshop)(I can demonstrated how the code works during workshop)

AllAll coupled channel models coupled channel models are based on solving Dyson-Schwinger are based on solving Dyson-Schwinger integral type equations, and they all have the same general structure:integral 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 is an is an isobar model isobar model wherewhere

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 is not calculated but effectively parameterized. interaction is not calculated but effectively parameterized.

Model is manifestly Model is manifestly unitary unitary and and analyticanalytic. .

channel-resonance channel-resonance mixing mixing matrix matrix

bare particle bare particle propagatorpropagatorchannel propagatorchannel propagator

Model assumptions: Model assumptions:

• isobar model (poles are introduced as intermediate isobar model (poles are introduced as intermediate resonant states called intermediate particles)resonant states called intermediate particles)

• background parameterization - background parameterization - meromorphic meromorphic functionfunction

• the form of imaginary part of the channel propagator the form of imaginary part of the channel propagator introduces proper channel cuts introduces proper channel cuts

2 2( ( ) )( ( ) )( )

4a

s M m s M mq s

s

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

Imaginary part of the channel propagator is Imaginary part of the channel propagator is defined as:defined as:

2 2( ( ) )( ( ) )( )

4a

s M m s M mq s

s

qqaa(s) is the meson-nucleon cms momentum:(s) is the meson-nucleon cms momentum:

The The analyticity ianalyticity iss manifestly manifestly imposed by calculating the channel imposed by calculating the channel propagator real part through the dispersion relation:propagator real part through the dispersion relation:

The The unitarityunitarity has been proven by Cutkosky. has been proven by Cutkosky.

The full solution is given as:The full solution is given as:

= = ijij

T

• we define the number of background poleswe define the number of background poles• we define the number of resonance poleswe define the number of resonance poles• we fit we fit ssi i ......... resonance mass......... resonance mass

ic ic ......... channel resonance mixing parameters ......... channel resonance mixing parameters

Final result: Final result: energy dependent partial wave T-matrices energy dependent partial wave T-matrices on the on the real axes real axes

Step 1: Step 1: Fitting procedureFitting procedure

Step 2: Step 2: Extracting resonance parameters – go into the Extracting resonance parameters – go into the complex energy plane complex energy plane (singularity (singularity structure of the obtained solution) structure of the obtained solution)

To find the position of poles of the matrix T(s) in the complex To find the position of poles of the matrix T(s) in the complex energy plane we have to find all zeroes of the denominator. So, energy plane we have to find all zeroes of the denominator. So, we solve the following equation: we solve the following equation:

How do we solve it?How do we solve it?

• when obtained from the fit, det Gwhen obtained from the fit, det G-1 -1 (s)(s) is a is a complexcomplex function of function of a a realreal argument s argument s

• we have towe have to

1.1. analytically continue analytically continue this function into the complex this function into the complex energy plane energy plane (observe that (observe that only only channel propagator channel propagator (s) (s) has to be analytically continuedhas to be analytically continued

2.2. find a complex zero find a complex zero ss00 of that function in the complex of that function in the complex

energy plane – energy plane – we do it numericallywe do it numerically

1.1. Analytical continuation of the channel propagator Analytical continuation of the channel propagator (s)(s)

a.a. Numerical integration Numerical integration ( (In old paperIn old paper))

b.b. Nowadays - Pietarinen expansionNowadays - Pietarinen expansion

We have constructed a function:We have constructed a function:

Observe that this is a complexObserve that this is a complex function of a complex argumentfunction of a complex argument for physical argument x! for physical argument x!

for x > xfor x > x0 0 xx00 – x is negative, and Z – x is negative, and ZII (x) is complex (x) is complex

How does it look in practice?How does it look in practice?

2.2. Finding a complex zeroFinding a complex zero

We did it numerically:We did it numerically:

• instead of calculating instead of calculating | det G | det G -1-1| | we have calculated we have calculated | det G || det G |• we made a 3D plot we made a 3D plot

| det G | = f (Re s, Im s)| det G | = f (Re s, Im s)

and numerically looked for the and numerically looked for the pointpoint of infinity of infinity of this function. of this function.

Mathematical operationalizationMathematical operationalization

Mass → Mass →

Width →Width →

Partial width →Partial width →

Stability of the procedureStability of the procedure

Stability of the procedure has been tested with respect Stability of the procedure has been tested with respect with different model assumptions of Zagreb CMB:with different model assumptions of Zagreb CMB:

1.1. Defining the inputDefining the input2.2. Form of the channel propagator (meson-resonance Form of the channel propagator (meson-resonance

vertex function)vertex function)• Inner partInner part• Asymptotic part Asymptotic part • Cut off parametersCut off parameters

3.3. Type of the background Type of the background 4.4. Number of channelsNumber of channels5.5. Mass of the effective channelMass of the effective channel

(To be given at the end of the talk if time permits....)(To be given at the end of the talk if time permits....)

ResultsResults

a.a. Use Zagreb CMB fits to analyze a particular PWD or PWAUse Zagreb CMB fits to analyze a particular PWD or PWA

b.b. Compare all PWA and PWD in Zagreb CMB in order to avoid Compare all PWA and PWD in Zagreb CMB in order to avoid systematic differences in analytic continuationsystematic differences in analytic continuation

Importance of inelastic channelsImportance of inelastic channels

Elastic channels only Elastic channels only are insufficient are insufficient to constrain to constrain all all T-matrix T-matrix poles, especially those which dominantly poles, especially those which dominantly couplecouple to inelastic to inelastic channels.channels.

In realityIn reality - - P P1111(1710) example(1710) example

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:

Use Zagreb CMB fits to analyze a particular PWD or PWAUse Zagreb CMB fits to analyze a particular PWD or PWA

I. I. Dubna – Mainz – Taipei Dubna – Mainz – Taipei (DMT) (DMT)

SS11 11 : : DMT model fits GWU/VPI single energy solutions , and obtains:DMT model fits GWU/VPI single energy solutions , and obtains:

They also fit They also fit ππNN→→ηηN SN S11 11

But we shall return to importance of elastic channels later.But we shall return to importance of elastic channels later.

Dilemma: Dilemma: How many dressed poles does one find in DMT functions?How many dressed poles does one find in DMT functions?

Facts of life:Facts of life:

• DMT model has 4 bare poles in SDMT model has 4 bare poles in S1111

•bare poles gets dressed, travel from the real axes into bare poles gets dressed, travel from the real axes into the complex energy plane, and there is no a priori way to the complex energy plane, and there is no a priori way to say where they endsay where they end•one needs either analytic continuation of DMT functions one needs either analytic continuation of DMT functions or some other pole search methodor some other pole search method•up to now DMT uses speed plot technique up to now DMT uses speed plot technique

But find only 3 polesBut find only 3 poles

We analyze their function with Zagreb CMB, and make We analyze their function with Zagreb CMB, and make a fit with 3 bare poles and 4 bare poles. a fit with 3 bare poles and 4 bare poles.

3R3R

4R4R

This is a good place to illustrate the difference in analytic This is a good place to illustrate the difference in analytic structure between CMB and other models. structure between CMB and other models.

At the same time this is a good place to illustrate the model At the same time this is a good place to illustrate the model dependence of bare parameters.dependence of bare parameters.

Both, Zagreb and DMT have the same structureBoth, Zagreb and DMT have the same structure full = bare + bare * interaction* fullfull = bare + bare * interaction* full

and this brings us to the self energy, and theand this brings us to the self energy, and the separation to the separation to the bare bare and and dresseddressed quantities. quantities.

I will illustrate that the I will illustrate that the way of way of separation separation

bare ↔bare ↔ dressed quantitiesdressed quantities

is different in DMT and Zagrebis different in DMT and Zagreb..

I take DMT amplitudes, fit them with Zagreb CMB,I take DMT amplitudes, fit them with Zagreb CMB, but fix bare but fix bare Zagreb Zagreb masses to DMT values:masses to DMT values:

MM1100 = 1559 MeV = 1559 MeV

MM2200 = 1727 MeV = 1727 MeV

MM3300 = 1803 MeV = 1803 MeV

MM4400= 2090 MeV = 2090 MeV

MM1100 = 1559 MeV = 1559 MeV

MM2200 = 1727 MeV = 1727 MeV

MM3300 = 1803 MeV = 1803 MeV

MM4400= 2090 MeV = 2090 MeV

MM1100 = 1506 MeV = 1506 MeV

MM2200 = 1650 MeV = 1650 MeV

MM3300 = 1853 MeV = 1853 MeV

MM4400= 2118 MeV = 2118 MeV

SS11 = 1600 + i 105 = 1600 + i 105

SS22 = 1680 + i 750 = 1680 + i 750

SS33 = 1790 + i 105 = 1790 + i 105

SS44 = 2140 + i 350 = 2140 + i 350

SS11 = 1501 + i 118 = 1501 + i 118

SS22 = 1640 + i 124 = 1640 + i 124

SS33 = 1921 + i 176 = 1921 + i 176

SS44 = 2129 + i 224 = 2129 + i 224

Best fitBest fit

ΧΧ22red red = 0.63= 0.63

ΧΧ22red red = 0.07= 0.07

Conclusion:Conclusion:

We support We support 4 bare pole solution, 4 bare pole solution, but find the but find the 44thth dressed pole too. dressed pole too.

We see three poles under 2 GeV and one above.We see three poles under 2 GeV and one above.

The latest DMT analysis goes beyond speed plot, uses analytic The latest DMT analysis goes beyond speed plot, uses analytic continuation, and says something different: continuation, and says something different: analytic continuation sees the analytic continuation sees the polepole where speed plot does where speed plot does not!not!

II. EBACII. EBAC

Original motivation: Original motivation: strong change in πN elastic PW when the ηN dastrong change in πN elastic PW when the ηN datta are included a are included

Compare Zagreb CMB fits with particular PWD or PWACompare Zagreb CMB fits with particular PWD or PWA

Diaz07 (Diaz07 (ππN elastic) PWAN elastic) PWA

Durand08 (Durand08 (ππN elastic + N elastic + ππN → N → ηηN) PWAN) PWA

Normalization of Normalization of ηηN PW?N PW?

Guided by:

In In publication publication we findwe find

We have analysed We have analysed

• Diaz07 (Diaz07 (ππN elastic) PWA N elastic) PWA

• Durand08 (Durand08 (ππN elastic + N elastic + ππN → N → ηηN) PWAN) PWA (as in publication)(as in publication)

•Durand08 (Durand08 (ππN elastic + N elastic + ππN → N → ηηN) PWA N) PWA (renormalized as suggested by D(renormalized as suggested by Dűring)űring)

Only πN elastic Diaz07 fittedOnly πN elastic Diaz07 fitted

Only Only ππN elastic Durand08 fittedN elastic Durand08 fitted

Renormalizatio suggested by Michael Dűring (December 2009)Renormalizatio suggested by Michael Dűring (December 2009)

Phase space factorPhase space factor

He replaced the PS factor for He replaced the PS factor for πN elastic πN elastic channelchannel

with the PS factor for with the PS factor for πNπN → → ηηN N channelchannel

So, he calculated the corerction factor N2/N1So, he calculated the corerction factor N2/N1

and obtainedand obtained

III. JuelichIII. Juelich

Compare Zagreb CMB fits with particular PWD or PWACompare Zagreb CMB fits with particular PWD or PWA

πN elastic πN elastic – 3R– 3R

πN elasticπN elastic + + πN→πN→ηηNN

Compare all PWA and PWD in Zagreb CMB in order to avoid Compare all PWA and PWD in Zagreb CMB in order to avoid systematic differences in analytic continuationsystematic differences in analytic continuation

πN→πN

KH80KH80

πN→πN + πN→ηN

ππN elastic N elastic

ππN elastic + N elastic + ππN → N → ηηNN

PiN-PiN-3R

PiN-PiN-4R

PiN-EtaN-3R

PiN-EtaN-4R

PiN-PiN-3R

PiN-EtaN-3R

PiN-PiN-4R

PiN-EtaN-4R

Stability of the procedureStability of the procedure

Stability of the procedure has been tested with respect Stability of the procedure has been tested with respect with different model assumptions of Zagreb CMB:with different model assumptions of Zagreb CMB:

1.1. Defining the inputDefining the input2.2. Form of the channel propagator (meson-resonance Form of the channel propagator (meson-resonance

vertex function)vertex function)• Inner partInner part• Asymptotic part Asymptotic part • Cut off parametersCut off parameters

3.3. Type of the background Type of the background 4.4. Number of channelsNumber of channels5.5. Mass of the effective channelMass of the effective channel

• Asymptotic part Asymptotic part

• Inner part Inner part (unchanged threshold behavior)(unchanged threshold behavior)

• Inner part Inner part (unchanged threshold behavior)(unchanged threshold behavior)

• Inner part Inner part (threshold behavior changed)(threshold behavior changed)

• Inner part Inner part (threshold behavior changed)(threshold behavior changed)