1. Multichannel Partial-Wave Analysis of Scattering Hongyu Zhang Tallahassee, FL October 12, 2005

2. Outline Introduction Database Formalism for Partial-Wave Analysis Fitting Procedures Results of Single-Energy Partial- Wave Analysis Summary

3. Introduction Since 1998, the Crystal Ball Collaboration at the BNL AGS has measured precise new data for several important reactions. These data have motivated a new partial-wave analysis (PWA) that is the subject of this research. Ultimate goal is to obtain more reliable information about properties of Λ  and Σ  resonances. This can be done by improvements in the experimental database and/or by improved partial-wave analysis techniques. GoalGoal

4. Introduction Our current knowledge of strangeness -1 hyperons is derived almost entirely from energy- dependent PWAs of scattering data. Energy-dependent PWAs assume a simple parametrization for the partial-wave amplitudes, which introduces a model-dependent bias and often results in a violation of unitarity of the S- matrix. One objective of our work is to reduce this bias as much as possible by carrying out a constrained energy-independent partial-wave analysis. Partial-Wave Analyses

5. Database Nuclear Physics B 6 (1968) 273-324 8 (1968) 233-264 20 (1970) 476-492 21 (1970) 15-76, 515-527 24 (1970) 417-440 29 (1971) 413-430 34 (1971) 41-70 67 (1973) 125-156 85 (1975) 289-310 90 (1975) 349-383 93 (1975) 189-216 96 (1975) 54-66 105 (1976) 189-221 Physical Review D 12 (1975) No. 1, 6-14 14 (1976) No. 1, 13-27 17 (1978) No. 9, 2226-2240 Numerical Data and Functional Relationships in Science and Technology Group I: Nuclear and Particle Physics, Vol. 12, Subvolume a Crystal Ball Collaboration (Private Communication) JournalsJournals

6. Database Channel P lab (MeV/c) d σ /d Ω P σ K-pK-p 281-1815 3,987585170 K0nK0n 281-1434 2,9130213 π0Λπ0Λ 436-1843 2,265128182 π+Σˉπ+Σˉ 436-1842 1,8670141 π0Σ0π0Σ0 436-1730 5017294 πˉ Σ + 436-1842 1,8760131 Total 15,02513,409785831 Momentum Range and Statistics

7. Formalism for Partial-Wave Analyses Unitarity Relations Previous Partial-Wave Analyses

8. Formalism for Partial-Wave Analyses Types of Unitarity Violation Observed: Unitarity Violation in Prior PWAs

9. Analysis Method Analysis Method Unitarization of selected “best” published amplitudes Unitarization of selected “best” published amplitudes Constrained single-energy fits of world data for: Constrained single-energy fits of world data for: Fitting Procedures

10. Constraints Constraints Small amplitudes (|T|<0.05) held fixed using unitarized solution Selected data bins of typically 30 MeV width Parameterize each amplitude in bin by: T(E)≈T(E 0 )+T’(E 0 )(E-E 0 ) where E is the CM energy of the data point in bin, E 0 is the center energy in bin, T(E 0 ) is the complex T-matrix amplitude at CM energy E 0, T’(E 0 ) is the “slope parameter” which is fixed at value from unitarized solution

11. Results of Single-Energy Partial-Wave Analysis

16. Summary What has been done: The available world database of d σ /d Ω, total cross sections, and polarization up to ~2 GeV, has been compiled, involving the reactions Initialized with a set of unitarized partial-wave amplitudes, after obtaining a reasonably smooth set of single-energy solutions for the amplitudes, an energy-dependent fit was carried out to ensure that the final results are consistent with unitarity. What remains to be done: Perform a unitarized fit based on our single-energy results Extract resonance parameters in a consistent manner for all channels

17. Acknowledgments D. Mark Manley, John Tulpan from Kent State University Crystal Ball Collaboration U. S. Department of Energy, grant DE-FG02-01ER41194 D. Mark Manley, John Tulpan from Kent State University Crystal Ball Collaboration U. S. Department of Energy, grant DE-FG02-01ER41194