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. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 2 / 33 Abstract model-independent extraction of the resonance parameters is enabled by imposing of the physical constraints to the scattering matrix.

3. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 3 / 33 Resonant Scattering strong peaks of the cross section of the meson-nucleon scattering: a manifestation of the resonance phenomena,strong peaks of the cross section of the meson-nucleon scattering: a manifestation of the resonance phenomena, resonances – excited baryons,resonances – excited baryons, baryon spectroscopy – obtaining of the resonance parameters (masses and widths) from the scattering data,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.at energies close to the baryon-resonance masses, the microscopic model (QCD) is still insolvable.

4. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 4 / 33 Howwouldoneobtainresonanceparameters Breit-Wigner parameterization Fit of the experimental data Excited baryon states ? ? Resonant parameters Dataset Experimental Dataset Data set Expansion Partial wave BW parameterization Fit to the partial waves Theoretical constraints + Parameterization Extraction P w a r v t e i s a l set Functions Parameterization of the partial waves Procedure Relevant analysis Legend: Topics covered here Datasets Procedures P w a r v t e i s a l

5. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 5 / 33 The Relevant Analyses What are relevant analyses Arndt FA02Arndt FA02 Manley MS92Manley MS92 Hoehler KH80Hoehler KH80 Cutkosky CMB (‘79)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),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,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 work properly, some “additional” requirements must be met.in order for many extraction methods to work properly, some “additional” requirements must be met.

6. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 6 / 33 It was (allegedly) easier in the old days... Transition matrix - TTransition matrix - T T = T R + T BT = T R + T B like adding of the Feynman diagramslike adding of the Feynman diagramsor Caley’s transform of unitary scattering matrix - KCaley’s transform of unitary scattering matrix - K K = K R + K BK = K R + K B the scattering matrix unitarity is conservedthe scattering matrix unitarity is conservedor Phase shift - Phase shift -   =  R +  B  =  R +  B addition of potentialsaddition of potentials

7. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 7 / 33... when people were using these simple resonant contributions... Resonant matrix T RResonant matrix T R T R = (  /2) / (M – W – i  /2)T R = (  /2) / (M – W – i  /2)or Resonant matrix K RResonant matrix K R K R = (  /2) / (M – W)K R = (  /2) / (M – W)or Resonant phase shift -  RResonant phase shift -  R  R  = arctan[(  /2) / (M – W)]  R  = arctan[(  /2) / (M – W)] 0.0 1.0 -0.5 0.5 M W/a.u. Im T R Re T R 

8. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 8 / 33... but, was it really? T-matrix addition T-matrix addition generally violates the S-matrix unitarity,generally violates the S-matrix unitarity, K-matrix addition K-matrix addition conserves unitarity, but the approach is not unique,conserves unitarity, but the approach is not unique, Phase shift additionPhase shift addition comes down to S-matrix multiplication (yet another model),comes down to S-matrix multiplication (yet another model), in multichannel cases, order of matrix multiplications is not defined (instead of the scalar  we have matrix , coming from S = e 2i  ).in multichannel cases, order of matrix multiplications is not defined (instead of the scalar  we have matrix , coming from S = e 2i  ).

9. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 9 / 33 What are, then, resonant parameters? Natural attempts Matrices T and K have poles:Matrices T and K have poles: T = K (I – i K) -1,T = K (I – i K) -1, K = T (I + i T) -1.K = T (I + i T) -1. S has common poles with T:S has common poles with T: S = I + 2 i T.S = I + 2 i T. What could be done with  ?What could be done with  ? In baryon spectroscopy Pole parameters:Pole parameters: pole of T matrix Breit-Wigner parameters:Breit-Wigner parameters: fit of the BW parameterization to the T (or K?) matrix

10. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 10 / 33 The Choices We Must Make: picking the resonance contribution there are many possible choices – how to pick the right one?there are many possible choices – how to pick the right one? it is much simpler if the scattering is:it is much simpler if the scattering is: single channel (all inelastic channels closed),single channel (all inelastic channels closed), single resonant (just one resonance contributes),single resonant (just one resonance contributes), with constant resonant “parameters”.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,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,The Assumption: energy dependent partial waves has been established properly, Now we must determine proper resonance parameters, and extract them.Now we must determine proper resonance parameters, and extract them.

11. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 11 / 33 To find physical criteria,To find physical criteria, To determine resonant parameters based on those criteria,To determine resonant parameters based on those criteria, To test (i.e. apply) method on well known example.To test (i.e. apply) method on well known example. CONCEPT Non-uniqueness of the extraction results Model dependence Scattered values of the resonant parameters Framework for the method(s) Extraction method Establishing the method(s) Application Testing of the method(s) Critical problem of baryon spectroscopy: Methods’ model dependence and non-uniqueness of the result Resonant pole Amplitude analyticity Time inversion symmetry Probability conservation Physical analysis (criteria) Extraction of the resonance parameters Comparison with initial results RESULTS Uniqueness, analyticity unitarity and symmetric Generalization: All matrices carry the same information Breit-Wigner HybridsSpeed plot Time delay Disagreements Point to analysis problems Physical approach Legitimate methods: K matrix pole (trace method) T matrix pole (regularization) Strong correlation of T-matrix trace and poles of K Physical criteria PROBLEM(S) OBJECTIVES APPROACH List of physical criteria: Non working methods: Choice: CMB approach Breit-Wigner K matrix Phase shift Hybrids T matrix Speed plot Time delay Main idea of the extraction methods

12. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 12 / 33 Physical criteria probabilities add up to one,probabilities add up to one, time-inversion invariance,time-inversion invariance, parity is conserved in QCD,parity is conserved in QCD, total spin is also conserved,total spin is also conserved, quantum and relativistic mechanics – occurrence of the propagatorsquantum and relativistic mechanics – occurrence of the propagators scattering matrix unitarity,scattering matrix unitarity, symmetrical scattering matrix,symmetrical scattering matrix, parity is a good quantum number,parity is a good quantum number, spin is a good quantum number,spin is a good quantum number, poles emerges.poles emerges. Physical conditions Scattering matrix

13. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 13 / 33 Infinite cross section?! Propagator GPropagator G G o = i (p 2 - m 2 + i  ) -1G o = i (p 2 - m 2 + i  ) -1 G (1) = G oG (1) = G o masses of products greater then the propagator mass – infinite contribution to cross section at some physical energy (simple pole)masses of products greater then the propagator mass – infinite contribution to cross section at some physical energy (simple pole) next order – loop contribution i next order – loop contribution i  G (2) = G o + G o i  G oG (2) = G o + G o i  G o even worse – pole is now second ordereven worse – pole is now second order

14. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 14 / 33 G o = i (p 2 – m 2 + i  ) -1 m 2  m 2 – i , p = sqrt(p 2 ) i  = i p  f  Re, p = sqrt(p 2 ) G = i (p 2 – m 2 + ip  f + Re) -1 Standard Breit-Wigner: Re = 0, f = 1, p = m. Flatte approximation: Re = 0, f = 1.

15. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 15 / 33 Scattering matrix unitarity scattering matrix is unitary so it can be diagonalized by some unitary matrix Uscattering matrix is unitary so it can be diagonalized by some unitary matrix U S = U + S D US = U + S D U S D is diagonal matrixS D is diagonal matrix S D =  s a E aS D =  s a E a matrix E a is a vector of orthonormal basis for diagonal matrix decompositionmatrix E a is a vector of orthonormal basis for diagonal matrix decomposition We define matrices  aWe define matrices  a  a = U + E a U  a = U + E a U S matrix expansionS matrix expansion S =   a s a  =   a e   i  a S =   a s a  =   a e   i  a Matrix  properties hermiticity (  a ) + =  ahermiticity (  a ) + =  a orthoidempotence  a  b =  a  aborthoidempotence  a  b =  a  ab completeness    a = Icompleteness    a = I trace Tr  a = 1trace Tr  a = 1

16. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 16 / 33 The rest of the physical demands Other physical conditions on matrix Other physical conditions on matrix   I J  I J  parity, spin and isospin conservation  I J  I J  parity, spin and isospin conservation  T =  time inversion invariance  T =  time inversion invariance  a  =  O T E a O  matrix has no poles on the real axis (orthogonal O instead of unitary U).  a  =  O T E a O  matrix has no poles on the real axis (orthogonal O instead of unitary U). Poles come from  a and other functions of  a :Poles come from  a and other functions of  a : sin  a e i  a  tan  a   e   i  asin  a e i  a  tan  a   e   i  a A new question:A new question: What is going on with  outside of the real axes?What is going on with  outside of the real axes?

17. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 17 / 33 Generalization of S-matrix “offshoots” by  matrices S-matrix “offshoots” matrices T, K i  may be derived from S matrix,matrices T, K i  may be derived from S matrix, matrices S, T, K i  carry the same information – what differs them is our capability to extract it,matrices S, T, K i  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) orthogonal matrices (O).what is interesting is the fact that matrices K, Im T i Re S are diagonalized by the same (real) orthogonal matrices (O).

18. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 18 / 33 Trace – channel-dependence elimination T =   a t a  Tr Tr  a = 1T =   a t a  Tr Tr  a = 1 Tr T =  t a  =  sin  a e i  a =t r + t b Tr K =  k a  =  tg  a = k r + k b Tr  =   a =   a =  r +  b Tr S =  s a  =  e   i  a = s r + s b Trace of the matrix  is 1 on the real axis: is it  (i.e. Tr  ) analytic function?is it  (i.e. Tr  ) analytic function? if it is, it will have value 1 in the vicinity of the real axis as well (in the resonant area);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 of phase shift , and will not care about the energy dependence of .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 of phase shift , and will not care about the energy dependence of .

19. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 19 / 33 Resonant parameter extraction: types of parameters S or T have pole (pole T) K has pole (pole K)  has “something”  /2 (?)  a is  /2 (pole K) tan(  a ) 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 poleT-matrix pole K-matrix poleK-matrix pole But there could be some other possibilities...

20. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 20 / 33 Up-to-date resonant parameters definitions T-matrix pole T R = r / (  – W) Speed plot |dT R /dW| = |r| / [(Re  – W) 2 +  Im  2 ] Time delay d  R /dW = (  /2) / [(M – W) 2 +  2 /4] Hybrids? K-matrix pole K R = (  /2) / (M – W) Breit-Wigner fit T R = (  /2) / (M – W – i  /2) Phase shift is  /2  R = arctan[(  /2) / (M – W)]

21. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 21 / 33 Poles of K and T matrices These are not (necessarily) parameterizations!

22. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 22 / 33 K-matrix Poles: The Recipe I K-matrix Poles: The Recipe I S. Ceci, A. Švarc, B. Zauner, D. M. Manley and S. Capstick, hep-ph/0611094v1

23. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 23 / 33 T-matrix Poles: Regularization Method channel opening resonance pole Almost ideal case: single resonance case parabolic behavior unstable region convergent region convergent segment

24. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 24 / 33 T-matrix Poles: The Recipe II S. Ceci, J. Stahov, A. Švarc, S. Watson and B. Zauner, hep-ph/060923v1 elastic channel opening continuum opening eta channel opening resonance pole N(1535) resonance pole N(1650) Nontrivial case of the N(1535) N(1535) convergent segment Small N(1535) convergence region Unstable regions

25. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 25 / 33 What is in fact speed plot? N = 1 Speed plot |dT R /dW| = |r| / [(Re  – W) 2 + (Im  ) 2 ]

26. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 26 / 33 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),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),it is generally considered as a nifty way of finding pole position (NSTAR 2004), many PWA groups use it (NSTAR 2007),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),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,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.many papers utilizing speed plot or time delay are already published, and some might be published as we speak.

27. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 27 / 33 Up-to-date resonant parameters definitions T-matrix pole T R = r / (  – W) Speed plot |dT R /dW| = |r| / [(Re  – W) 2 +  Im  2 ] Time delay d  R /dW = (  /2) / [(M – W) 2 +  2 /4] Hybrids? K-matrix pole K R = (  /2) / (M – W) Breit-Wigner fit T R = (  /2) / (M – W – i  /2) Phase shift is  /2  R = arctan[(  /2) / (M – W)] T-matrix pole: first order approximation K-matrix pole Model dependent

28. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 28 / 33 Which came first ? K-matrix and its polesT-matrix and its poles

29. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 29 / 33 Choice of the analysis Choice of the analysis CMB unitary (S) analytic analytic multiresonance multiresonance coupled channel coupled channel experimentally obtained theory-produced functions parameterization by elementary functions functions from other sources

30. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 30 / 33 K-matrix pole parameters – values It seems like there is some relation between BW and K- matrix pole parameters?

31. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 31 / 33 Correlation – K-poles and T-trace Peak Zero

32. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 32 / 33 T-matrix poles: comparison of regularization and analytic continuation. Interesting is that strong discrepancies occur at the same places as in the case of the K-matrix poles.

33. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 33 / 33 Some results of this research Physics: two extraction methods are developed (regularization method i trace method),two extraction methods are developed (regularization method i trace method), correlation between pole positions of K matrix and trace of matrix T is quite strong (but also unexpected),correlation between pole positions of K matrix and trace of matrix T is quite strong (but also unexpected), stronger departures from this correlation indicated (practically always) to known issues of the original analysis,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,we expect that this could be improved by addition of a few additional important inelastic channels, projector matrices  most likely do not have resonant poles.projector matrices  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),a method is developed that determines simple poles or zeros of complex functions (verified on various independent cases), projector matrices  are introduced as a basis for expansion of the normal matrices.projector matrices  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),origin and misinterpretations are given of the speed plot method (similarly for the time delay), critics is drawn on the model-dependent methods (like various BW parameterizations and hybrid approaches)critics is drawn on the model-dependent methods (like various BW parameterizations and hybrid approaches) most methods are developed for narrow resonances – in baryon physics they are simply not narrow enough.most methods are developed for narrow resonances – in baryon physics they are simply not narrow enough.

35. Helsinki, June 27 (2007)Saša Ceci, The 4th International PWA Workshop 35 / 33 OK, but who really cares what the speed plot in fact is?...