1. A Primer on Partial Wave Analysis in hadron spectroscopy Charm 2006 International Conference on Tau-Charm Physics Beijing, June 5-7 Klaus Peters IKF, JWGU Frankfurt und KP3, GSI Darmstadt

2. 2 Klaus Peters - PWA Primer Overview What do we need to talk about ? Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues

3. 3 Klaus Peters - PWA Primer Overview – Introduction and Concepts What do we need to talk about ? Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues Goals Wave Approach Isobar-Model Level of Detail

4. 4 Klaus Peters - PWA Primer Overview – Spin Formalisms What do we need to talk about ? Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues Overview Zemach Formalism Canonical Formalism Helicity Formalism Moments Analysis

5. 5 Klaus Peters - PWA Primer Overview - Dynamical Functions What do we need to talk about ? Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues Breit-Wigner S-/T-Matrix K-Matrix P/Q-Vector N/D-Method Barrier Factors Interpretation

6. 6 Klaus Peters - PWA Primer Overview – Technical Issues / Fitting What do we need to talk about ? Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues Coding Amplitudes Speed is an Issue Fitting Methods Caveats FAQ

7. 7 Klaus Peters - PWA Primer (non-theorist) References hep-ex/0410014: 18p., D. Asner, Charm Dalitz Plot Analysis Formalism and Results Expanded version of review in "Review of Particle Physics", S. Eidelman et al., Phys. Lett. B 592, 1 (2004) hep-ph/0412069: 62p., K. Peters, A Primer on Partial Wave Analysis Lectures given at International Enrico Fermi School of Physics, Varenna, Italy, 6-16 Jul 2004. published  Varenna 2004, Hadron physics p. 451-514 Charm 2006 D. Asner M. Pappagallo M. Pennington

8. 8 Klaus Peters - PWA Primer Overview – Technical Issues / Fitting What do we need to talk about ? Introduction and Concepts Spin Formalisms Dynamical Functions Technical Issues

9. 9 Klaus Peters - PWA Primer What is the mission ? Particle physics at small distances is well understood One Boson Exchange, Heavy Quark Limits This is not true at large distances Hadronization, Light mesons are barely understood compared to their abundance Understanding interaction/dynamics of light hadrons will improve our knowledge about non-perturbative QCD parameterizations will give provide toolkit to analyze heavy quark processes thus an important tool also for precise standard model tests We need Appropriate parameterizations for the multi-particle phase space A translation from the parameterizations to effective degrees of freedom for a deeper understanding of QCD

10. 10 Klaus Peters - PWA Primer Goal For whatever you need the parameterization of the n-Particle phase space It contains the static properties of the unstable (resonant) particles within the decay chain like mass width spin and parities as well as properties of the initial state and some constraints from the experimental setup/measurement The main problem is, you don‘t need just a good description, you need the right one Many solutions may look alike but only one is right and they differ strongly in the phases involved

11. 11 Klaus Peters - PWA Primer Intermediate State Mixing Many states may contribute to a final state not only ones with well defined (already measured) properties not only expected ones Many mixing parameters are poorly known K-phases SU(3) phases In addition also D/S mixing (b 1, a 1 decays)

12. 12 Klaus Peters - PWA Primer n-Particle Phase space, n=3 2 Observables From four vectors12 Conservation laws-4 Meson masses-3 Free rotation-3 Σ2 Usual choice Invariant mass m 12 Invariant mass m 13 π3π3 π2π2 p π1π1 Dalitz plot

13. 13 Klaus Peters - PWA Primer J/ψ  π + π - π 0 Angular distributions are easily seen in the Dalitz plot cosθ 0+1

14. 14 Klaus Peters - PWA Primer It’s All a Question of Statistics... pp  3 0 with 100 events

15. 15 Klaus Peters - PWA Primer It’s All a Question of Statistics...... pp  3 0 with 100 events 1000 events

16. 16 Klaus Peters - PWA Primer It’s All a Question of Statistics......... pp  3 0 with 100 events 1000 events 10000 events

17. 17 Klaus Peters - PWA Primer It’s All a Question of Statistics............ pp  3 0 with 100 events 1000 events 10000 events 100000 events

18. 18 Klaus Peters - PWA Primer Introducing Partial Waves Schrödinger‘s Equation Angular Amplitude Dynamic Amplitude

19. 19 Klaus Peters - PWA Primer Spin Formalisms – on overview Tensor formalisms in non-relativistic (Zemach) or covariant form Fast computation, simple for small L and S Spin-projection formalisms where a quantization axis is chosen and proper rotations are used to define a two-body decay Efficient formalisms, even large L and S easy to handle Formalisms based on Lorentz invariants (Rarita-Schwinger) where each operator is constructed from Mandelstam variables only Elegant, but extremely difficult for large L and S

20. 20 Klaus Peters - PWA Primer Argand Plot

21. 21 Klaus Peters - PWA Primer Standard Breit-Wigner Full circle in the Argand Plot Phase motion from 0 to π Intensity I=ΨΨ * Phase δSpeed dφ/dm Argand Plot

22. 22 Klaus Peters - PWA Primer Breit-Wigner in the Real World e + e - ππ m ππ ρ-ωρ-ω

23. 23 Klaus Peters - PWA Primer Dynamical Functions are Complicated Search for resonance enhancements is a major tool in meson spectroscopy The Breit-Wigner Formula was derived for a single resonance appearing in a single channel But: Nature is more complicated Resonances decay into several channels Several resonances appear within the same channel Thresholds distort line shapes due to available phase space A more general approach is needed for a detailed understanding

24. 24 Klaus Peters - PWA Primer Isobar Model Generalization construct any many-body system as a tree of subsequent two-body decays the overall process is dominated by two-body processes the two-body systems behave identical in each reaction different initial states may interfere We need need two-body “spin”-algebra various formalisms need two-body scattering formalism final state interaction, e.g. Breit-Wigner Isobar

25. 25 Klaus Peters - PWA Primer K-Matrix Definition T is n x n matrix representing n incoming and n outgoing channel If the matrix K is a real and symmetric also n x n then the T is unitary by construction

26. 26 Klaus Peters - PWA Primer Example: 1x2 K-Matrix Nearby Poles Two nearby poles (1.27 and 1.5 GeV/c 2 ) show nicely the effect of unitarization 2 BW K-Matrix

27. 27 Klaus Peters - PWA Primer Flatté Example a 0 (980) decaying into πη and KK BW πη Flatte πη Flatte KK Intensity I=ΨΨ * Phase δ Real Part Argand Plot

28. 28 Klaus Peters - PWA Primer Example: K-Matrix Parametrizations Au, Morgan and Pennington (1987) Amsler et al. (1995) Anisovich and Sarantsev (2003)

29. 29 Klaus Peters - PWA Primer Experimental Techniques Scattering Experiments πN - N* measurement πN - meson spectroscopy E818, E852 @ AGS, GAMS pp meson threshold production Wasa @ Celsius, COSY pp or πp in the central region WA76, WA91, WA102 γN – photo production Cebaf, Mami, Elsa, Graal “At-rest” Experiments pN @ rest at LEAR Asterix, Obelix, Crystal Barrel J/ψ decays MarkIII,DM2,BES,CLEO-c ф(1020) decays Kloe @ Dafne, VEPP D and D s decays FNAL, Babar, Belle

30. 30 Klaus Peters - PWA Primer Experimental Techniques Scattering Experiments partial waves decomposition  via moment analysis systematic studies to limit #waves dynamics appear as amplitude variations resonance parameters from fits to amplitudes “At-rest” Experiments ad-hoc introduction of waves ad-hoc introduction of dynamic amplitudes (“resonances”) systematic studies to limit #waves and #resonances resonance parameters appear as fit parameters

31. 31 Klaus Peters - PWA Primer Experimental Techniques Scattering Experiments exchange model needed ad-hoc intermediate resonances  parameters fixed for wave decomposition “At-rest” Experiments independent of production model intermediate resonances treated  identically to final state resonances crossing bands may provide high resolution interferometer

32. 32 Klaus Peters - PWA Primer Moments Analysis Consider reaction Total differential cross section expand H leading to

33. 33 Klaus Peters - PWA Primer E852 f 1 π and b 1 π blue one 1 -+ pole black two poles  2 =70.6/47=1.5 E852

34. 34 Klaus Peters - PWA Primer Fit for D 0 K s π + π - see M. Pappagallo, this conf.

35. 35 Klaus Peters - PWA Primer Overlapping band usually make it very difficult to do a moment analysis to get an impression on the wave content

36. 36 Klaus Peters - PWA Primer Momentum Analysis in a Dalitz Plot In some cases it‘s possible if no sharp bands overlap see M. Pappagallo, this conf.

37. 37 Klaus Peters - PWA Primer Technical Aspects Initial composition for fits ? How to treat (non-peaking) background ? How to identifiy the best fit ? Numerical aspects for complex dynamical functions ? Speed Issues, Precision of MC phase space integrals ! Systematic aspects of scalar waves !

38. 38 Klaus Peters - PWA Primer Summary Partial Wave decompostion and proper treatment of resonances (dynamics, spins, parities) has become extremely important in charm and beauty physics Required for a proper understanding multibody D (S) decays In particular Measurement of γ/Φ 3 via D  K S ππ as interferometer charm-Mixing from time-dependent Dalitzplot fits For this purpose: one needs a correct parameterization which yield correct phases !