Model-independent R esonance P arameters

1. Model-independent Resonance Parameters Saša Ceci, Alfred Švarc, Branimir Zauner Ruđer Bošković Institute with special thanks to Mike Sadler, Shon Watson, Jugoslav Stahov, Pekko Metsa, Mark Manley and Simon Capstick Helsinki 2007 The 4th International PWA Workshop

2. Abstract model-independent extraction of the resonance parameters is enabled by imposing of the physical constraints to the scattering matrix. Saša Ceci, The 4th International PWA Workshop

3. Resonant Scattering • strong peaks of the cross section of the meson-nucleon scattering: a manifestation of the resonance phenomena, • resonances – excited baryons, • baryon spectroscopy – obtaining of the resonance parameters (masses and widths) from the scattering data, • at energies close to the baryon-resonance masses, the microscopic model (QCD) is still insolvable. Saša Ceci, The 4th International PWA Workshop

4. Breit - Wigner parameterization Fit of the experimental data BW parameterization P w Fit to the partial waves a a r v te Partial wave P w ? ? i s Expansion a a Excited baryon states Experimental a Dataset Resonant parameters r v Dataset Theoretical l te constraints i s + a Parameterization l Extraction Data set Functions Parameterization set of the partial waves Relevant analysis Topics Datasets Legend: Procedures covered here Procedure How would one obtain resonance parameters Saša Ceci, The 4th International PWA Workshop

5. What are relevant analyses Arndt FA02 Manley MS92 Hoehler KH80 Cutkosky CMB (‘79) Origin: Review of Particle Physics (PDG). Getting spectra can be difficult partial-wave functions are obtained from the numerical calculations (DR, expansions on nontrivial functions), there are too many models, and it is not clear which one (and in which cases) should be used, in order for many extraction methods to workproperly, some “additional” requirements must be met. The Relevant Analyses Saša Ceci, The 4th International PWA Workshop

6. It was (allegedly) easier in the old days ... • Transition matrix - T • T = TR + TB • like adding of the Feynman diagrams or • Caley’stransform of unitary scattering matrix - K • K = KR + KB • the scattering matrix unitarity is conserved or • Phase shift - d • d = dR + dB • addition of potentials Saša Ceci, The 4th International PWA Workshop

7. ... when people were using these simple resonant contributions... • Resonant matrix TR • TR = (G/2) / (M – W – i G/2) or • Resonant matrix KR • KR = (G/2) / (M – W) or • Resonant phase shift - dR • dR= arctan[(G/2) / (M – W)] 1.0 0.5 G Im TR 0.0 M Re TR -0.5 W/a.u. Saša Ceci, The 4th International PWA Workshop

8. ... but, was it really? • T-matrix addition • generally violates the S-matrix unitarity, • K-matrix addition • conserves unitarity, but the approach is not unique, • Phase shift addition • comes down to S-matrix multiplication (yet another model), • in multichannel cases, order of matrix multiplications is not defined (instead of the scalar d, we have matrix D, coming from S = e2iD). Saša Ceci, The 4th International PWA Workshop

9. Natural attempts Matrices T and K have poles: T = K (I – i K)-1, K = T (I + i T)-1. S has common poles with T: S = I + 2 i T. What could be done with d? In baryon spectroscopy Pole parameters: pole of T matrix Breit-Wigner parameters: fit of the BW parameterization to the T (or K?) matrix What are, then, resonant parameters? Saša Ceci, The 4th International PWA Workshop

10. The Choices We Must Make: picking the resonance contribution • there are many possible choices – how to pick the right one? • it is much simpler if the scattering is: • single channel (all inelastic channels closed), • single resonant (just one resonance contributes), • with constant resonant “parameters”. • generalization to multichannel and multiresonance situations, with nontrivial energy dependence of background contributions and resonant parameters calls for quite elaborate approaches, • The Assumption:energy dependent partial waves has been established properly, • Now we must determine proper resonance parameters, and extractthem. Saša Ceci, The 4th International PWA Workshop

11. CONCEPT Main idea Model dependence Non-uniqueness Scattered values of the extraction methods of the extraction results of the resonant parameters PROBLEM(S) Critical problem of baryon spectroscopy: • To find physical criteria, • To determine resonant parameters based on those criteria, • To test(i.e. apply) method on well known example. Methods’ model dependence and non-uniqueness of the result Physical criteria Extraction method Application OBJECTIVES Framework for the method(s) Establishing the method(s) Testing of the method(s) Physical analysis (criteria) Resonant pole T matrix Breit-Wigner Extraction of the resonance K matrix Speed plot Amplitude analyticity APPROACH parameters Phase shift Time delay Time inversion symmetry Comparison with initial Hybrids Probability conservation results Choice: CMB approach Non working methods: List of physical criteria: Physical approach Uniqueness, analyticity Breit-Wigner Time delay unitarity and symmetric Speed plot Hybrids Disagreements RESULTS Point to analysis problems Generalization: Legitimate methods: All matrices carry the same information Strong correlation of K matrix pole (trace method) T matrix pole (regularization) T-matrix trace and poles of K Saša Ceci, The 4th International PWA Workshop

12. probabilities add up to one, time-inversion invariance, parity is conserved in QCD, total spin is also conserved, quantum and relativistic mechanics – occurrence of the propagators scattering matrix unitarity, symmetrical scattering matrix, parity is a good quantum number, spin is a good quantum number, poles emerges. Physical criteria Scattering matrix Physical conditions Saša Ceci, The 4th International PWA Workshop

13. Propagator G Go = i (p2 - m2 + ie)-1 G(1) = Go masses of products greater then the propagator mass – infinite contributionto cross section at some physical energy (simple pole) next order – loop contribution iS G(2)= Go + GoiS Go even worse – pole is now second order Infinite cross section?! Saša Ceci, The 4th International PWA Workshop

14. Go = i (p2 – m2 + ie)-1 m2-> m2 – iS iS = ipGf + Re, p = sqrt(p2) G = i (p2 – m2 + ipGf + Re)-1 Standard Breit-Wigner: Re = 0, f = 1, p = m. Flatte approximation: Re = 0, f = 1. Saša Ceci, The 4th International PWA Workshop

15. scattering matrix is unitary so it can be diagonalized by some unitary matrix U S = U+ SD U SDis diagonal matrix SD = S sa Ea matrix Eais a vector of orthonormal basis for diagonal matrix decomposition We define matricesca ca = U+ EaU S matrix expansion S =Sca sa= Sca e2ida Matrix c properties hermiticity (ca)+ = ca orthoidempotence ca cb = ca dab completeness Sca = I trace Tr ca = 1 Scattering matrix unitarity Saša Ceci, The 4th International PWA Workshop

16. The rest of the physical demands • Other physical conditions on matrix c: • cIJp , dIJp parity, spin and isospin conservation • cT = c time inversion invariance • ca=OT Ea O c matrix has no poles on the real axis (orthogonal O instead of unitary U). • Poles come from da and other functions of da: • sin da eida, tan da, e2ida • A new question: • What is going on with c outside of the real axes? Saša Ceci, The 4th International PWA Workshop

17. Generalizationof S-matrix “offshoots” bycmatrices S-matrix “offshoots” • matrices T, K i D may be derived from S matrix, • matrices S, T, K i D carry the same information – what differs them is our capability to extract it, • what is interesting is the fact that matrices K, Im T i Re S are diagonalized by the same (real) orthogonalmatrices (O). Saša Ceci, The 4th International PWA Workshop

18. Trace – channel-dependence elimination • T = Sca ta/ Tr Tr ca = 1 Tr T = S ta= Ssin da eida= tr + tb Tr K = S ka= S tg da = kr + kb Tr D = Sda= Sda = dr + db Tr S = S sa= S e2ida = sr + sb Trace of the matrixcis 1 on the real axis: • is itc (i.e. Tr c) analytic function? • if it is, it will have value 1 in the vicinity of the real axis as well (in the resonant area); • then we can say this: Pole positions of the T matrix, as well as those of the K matrix, will depend on the energy dependence ofphase shift d, and will not care about the energy dependence ofc. Saša Ceci, The 4th International PWA Workshop

19. S or T have pole (pole T) K has pole (pole K) D has “something” p/2 (?) da is p/2 (pole K) tan(da) has pole (pole K) speed plot shows peak (“pole T”) time delay shows peak (“pole T”) imaginary part of T matrix peak (BW?) Basically, two different ways: T-matrix pole K-matrix pole But there could be some other possibilities ... Resonant parameter extraction: types of parameters Saša Ceci, The 4th International PWA Workshop

20. T-matrix pole TR = r / (m – W) Speed plot |dTR/dW| = |r| / [(Re m – W)2 + (Im m)2] Time delay ddR/dW = (G/2) / [(M – W)2 + G2/4] Hybrids ? K-matrix pole KR = (G/2) / (M – W) Breit-Wigner fit TR = (G/2) / (M – W – iG/2) Phase shift is p/2 dR = arctan[(G/2) / (M – W)] Up-to-dateresonant parameters definitions Saša Ceci, The 4th International PWA Workshop

21. Poles of K and T matrices These are not (necessarily) parameterizations! Saša Ceci, The 4th International PWA Workshop

22. K-matrix Poles: The Recipe I S. Ceci, A. Švarc, B. Zauner, D. M. Manley and S. Capstick, hep-ph/0611094v1 Saša Ceci, The 4th International PWA Workshop

23. T-matrix Poles: Regularization Method Almost ideal case: single resonance case channel opening resonance pole convergent segment parabolic behavior unstable region convergent region Saša Ceci, The 4th International PWA Workshop

24. T-matrix Poles: The Recipe II Nontrivial case of the N(1535) elastic channel opening continuum opening eta channel opening resonance pole N(1535) resonance pole N(1650) N(1535) convergent segment Unstable regions Small N(1535) convergence region S. Ceci, J. Stahov, A. Švarc, S. Watson and B. Zauner, hep-ph/060923v1 Saša Ceci, The 4th International PWA Workshop

25. What isin fact speed plot? N =1 Speed plot |dTR/dW| = |r| / [(Re m – W)2 + (Im m)2] Saša Ceci, The 4th International PWA Workshop

26. OK, but who does really care what the speed plot in fact is? • KH80 pole parameters are obtained by using of the speed plot (the relevant values), • it is generally considered as a nifty way of finding pole position (NSTAR 2004), • many PWA groups use it (NSTAR 2007), • Excited Baryon Analysis Center (EBAC) at JLab in fact wants to study the validity of the speed plot and time delay (this years whitepaper), • it is considered as a model-independent extraction method, • many papers utilizing speed plot or time delay are already published, and some might be published as we speak. Saša Ceci, The 4th International PWA Workshop

27. T-matrix pole TR = r / (m – W) Speed plot |dTR/dW| = |r| / [(Re m – W)2 + (Im m)2] Time delay ddR/dW = (G/2) / [(M – W)2 + G2/4] Hybrids ? K-matrix pole KR = (G/2) / (M – W) Breit-Wigner fit TR = (G/2) / (M – W – iG/2) Phase shift is p/2 dR = arctan[(G/2) / (M – W)] Up-to-dateresonant parameters definitions T-matrix pole: first order approximation Model dependent K-matrix pole T-matrix pole: first order approximation Model dependent Saša Ceci, The 4th International PWA Workshop

28. K-matrixand its poles T-matrixandits poles Which came first ? Saša Ceci, The 4th International PWA Workshop

29. CMB • unitary (S) • analytic • multiresonance • coupled channel Choice of the analysis experimentally obtained theory-produced functions parameterization by elementary functions functions from other sources Saša Ceci, The 4th International PWA Workshop

30. K-matrix pole parameters –values It seems like there is some relation between BW andK-matrix pole parameters? Saša Ceci, The 4th International PWA Workshop

31. Peak Zero Peak Zero Correlation – K-poles and T-trace Saša Ceci, The 4th International PWA Workshop

32. Interesting is that strong discrepancies occurat the same placesasin the case of the K-matrix poles. T-matrix poles: comparison ofregularization andanalytic continuation. Saša Ceci, The 4th International PWA Workshop

33. Physics: two extraction methods are developed (regularization method i trace method), correlation between pole positions of K matrix and trace of matrix T is quitestrong (but also unexpected), stronger departures from this correlation indicated (practically always) to known issues of the original analysis, we expect that this could be improved by addition of a few additional important inelastic channels, projector matrices c most likely do not have resonant poles. Mathematics: a method is developed that determines simple poles or zeros of complex functions (verified on various independent cases), projector matricesc are introduced as a basis for expansion of the normal matrices. Influence to standard approaches: origin and misinterpretations are given of the speed plot method (similarly for the time delay), critics is drawn on the model-dependentmethods (like various BW parameterizations and hybrid approaches) most methods are developed for narrow resonances –in baryon physics they are simply not narrow enough. Some results of this research Saša Ceci, The 4th International PWA Workshop

35. ... ... OK, but who really cares what the speed plot in fact is? Saša Ceci, The 4th International PWA Workshop