Poles of PWDataand PWAmplitudesin Zagreb model A. Švarc, S. Ceci, B. Zauner Rudjer Bošković Institute, Zagreb, Croatia M. Hadžimehmedović, H. Osmanović, J. Stahov Univerzityof Tuzla, TuzlaBosniaandHerzegovina
How do I see what is our main aim? Experiment Quarks Matching point ? bound states structures resonances
Höhler – LandoltBernstein 1984. Burkert – Lee 2004. Höhler – LandoltBernstein 1984. Svarc 2004 Ceci, Svarc, Zauner 2005.
Difference between PWD and PWA Höhler – LandoltBernstein 1984. explicit analytic form introduced • Phenomenological T-matrix • CMU-LBL • Zagreb • Argonne-Pittsburgh • effective Lagrangian • EBAC • Juelich • Dubna-Mainz-Taipei (DMT) • Giessen • Chew-Mandelstam K-matrix • GWU/VPI PWD PWA
How do I see what is our main aim? Experiment Quarks Breit-Wigner parameters bound states Pole parameters structures resonances Phase shifts
What is “better”: • Breit-Wigner parameters • or • Pole parameters The advantages and drawbacks
Breit – Wigner parameters: • Advantages: • defined on the real axes • simple to calculate • Drawbacks: • dependence on the choice of field variables • model dependent (background definition) Harry Lee BRAG 2001
Pole parameters • Advantages • invariant with respect to the choice of field variables • model independent • less model dependent • Drawbacks • hard to get because they lie in the complex energy plane Why?
Extractionof Breit-Wignerparameters The dependence upon background parameterization is a well know fact, but nothing has been done in PDG yet. PDG makes an average of all BW values, regardless of the way background has been introduced. This introduces an additional systematic error.
Suggestion by Harry Lee: Suggestion by Lothar Tiator: We all know that BW positions and parameters are not well defined but many people believe that within some uncertainty they can be given, and are very useful. Therefore I also tend to stick with them and would propose to keep them in PDG in the future. But only if we can give some proper definition and methods for extraction.
Extractionof pole parameters • Each PWA has a specific assumption on the analytic form. • Idea: • Let us use CMB formalism to analyze available PWD and PWAusingone andthe same analyticform. • We propose: • To use new PWD or PWA in addition to the existing ones and look for the shift of poles (shift of present ones, appearance of the new ones) • To use ONE analytic form (Zagreb CMB) for ALL EXISTING PWA and PWD, and: • extract poles from a particular PWD or PWA using Zagreb CMB and compare the outcome with the original result • compare agreement of poles of ALL EXISTING PWA and PWD in order to avoid systematic error because of differences in analytic forms.
Warning: When we analyze particularPWA, we do not say that we shall exactly reproduce pole positions given by thatparticular model. Results may differ, and the difference will show the significance of analytic form chosen to represent on-energy shell data. So, we are not checking if the pole positions in a certain model are correctly extracted, but rather seting up the way how to quantify the comparison of different curves.
1. … to use new PWD or PWA .... Initial attempts ...
In details repeated at NSTAR2005 - Tallahassee Technical problems on Zagreb side......
Using CMB to analyze a world collection of PWD and PWA and eliminate model assumptions on analytic form • Formulated at BRAG2007 • Research in progress REMARK Technical problems in Zagreb code are now eliminated. Code is running under LINUX. It is transferable from machine to machine, and is an open source code. Adjustments and improvements can be done. (I can demonstrated how the code works during workshop)
CMB coupled-channel model • 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
Carnagie-Melon-Berkely (CMB) model is anisobar model where 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. Model is manifestly unitary and analytic. bare particle propagator channel-resonance mixing matrix channel propagator
Model assumptions: • isobar model (poles are introduced as intermediate resonant states called intermediate particles) • background parameterization - meromorphic function • the form of imaginary part of the channel propagator introduces proper channel cuts Imaginary part of the channel propagator is defined as: where qa(s) is the meson-nucleon cms momentum:
The analyticity is manifestly imposed by calculating the channel propagator real part through the dispersion relation: qa(s) is the meson-nucleon cms momentum: The unitarity has been proven by Cutkosky.
The full solution is given as: = ij T
Step 1: Fitting procedure • we define the number of background poles • we define the number of resonance poles • we fit • si......... resonance mass • ic ......... channel resonance mixing parameters Final result: energy dependent partial wave T-matrices on the real axes Step 2: Extracting resonance parameters – go into the complex energy plane (singularity structure of the obtained solution)
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, we solve the following equation:
How do we solve it? • when obtained from the fit, det G-1 (s)is a complex function of a real argument s • we have to • analytically continue this function into the complex energy plane (observe that only channel propagator (s) has to be analytically continued • find a complex zero s0 of that function in the complex energy plane – we do it numerically
Analytical continuation of the channel propagator (s) • Numerical integration (In old paper) • Nowadays - Pietarinen expansion We have constructed a function: Observe that this is a complex function of a complex argument for physical argument x! for x > x0 x0 – x is negative, and ZI (x) is complex
Finding a complex zero • We did it numerically: • instead of calculating | det G -1| we have calculated | det G | • we made a 3D plot • | det G | = f (Re s, Im s) • and numerically looked for the • pointof infinity of this function.
Mass → Width → Partial width →
Stability of the procedure • Stability of the procedure has been tested with respect with different model assumptions of Zagreb CMB: • Defining the input • Form of the channel propagator (meson-resonance vertex function) • Inner part • Asymptotic part • Cut off parameters • Type of the background • Number of channels • Mass of the effective channel (To be given at the end of the talk if time permits....)
Results Use Zagreb CMB fits to analyze a particular PWD or PWA Compare all PWA and PWD in Zagreb CMB in order to avoid systematic differences in analytic continuation
Importance of inelastic channels Elastic channels only are insufficient to constrain all T-matrix poles, especially those which dominantly couple to inelastic channels.
We use: • CMB model for 3 channels: • p N, h N, and dummy channel p2N • p N elastic T matrices , PDG: SES Ar06 • p N¨h N T matrices, PDG:Batinic 95 We fit: πNelastic only p N¨h N only both channels
Use Zagreb CMB fits to analyze a particular PWD or PWA I. Dubna – Mainz – Taipei (DMT) S11 : DMT model fits GWU/VPI single energy solutions , and obtains:
TheyalsofitπN→ηN S11 But we shall return to importance of elastic channels later.
Dilemma: How many dressed poles does one find in DMT functions? • Facts of life: • DMT model has 4 bare poles in S11 • bare poles gets dressed, travel from the real axes into the complex energy plane, and there is no a priori way to say where they end • one needs either analytic continuation of DMT functions or some other pole search method • up to now DMT uses speed plot technique
We analyze their function with Zagreb CMB, and make a fit with 3 bare poles and 4 bare poles.