Multichannel Partial-Wave Multichannel Partial-Wave AnalysisAnalysis

of Scatteringof Scattering

Hongyu Zhang

Tallahassee, FLOctober 12, 2005

NK

OutlineOutline

• Introduction• Database• Formalism for Partial-Wave Analysis• Fitting Procedures• Results of Single-Energy Partial-

Wave Analysis• Summary

IntroductionIntroduction

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.

NK

GoalGoalGoalGoal

IntroductionIntroduction

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.

NK

Partial-Wave AnalysesPartial-Wave AnalysesPartial-Wave AnalysesPartial-Wave Analyses

DatabaseDatabase

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)

JournalsJournalsJournalsJournals

DatabaseDatabase

Channel Plab (MeV/c) dσ/dΩ P σ

K-p 281-1815281-1815 3,987 585 170

K0n 281-1434281-1434 2,913 0 213

π0Λ 436-1843436-1843 2,265 128 182

π+Σˉ 436-1842436-1842 1,867 0 141

π0Σ0 436-1730436-1730 501 72 94

πˉΣ+ 436-1842436-1842 1,876 0 131Total

15,02513,40

9785 831

Momentum Range and StatisticsMomentum Range and StatisticsMomentum Range and StatisticsMomentum Range and Statistics

Formalism for Partial-Wave AnalysesFormalism for Partial-Wave Analyses

Unitarity Relations

Previous Partial-Wave Analyses

Formalism for Partial-Wave AnalysesFormalism for Partial-Wave Analyses

Types of Unitarity Violation Observed:

Unitarity Violation in Prior PWAsUnitarity Violation in Prior PWAsUnitarity Violation in Prior PWAsUnitarity Violation in Prior PWAs

Analysis MethodAnalysis Method

• Unitarization of selected “best” Unitarization of selected “best” published amplitudespublished amplitudes

• Constrained single-energy fits of world Constrained single-energy fits of world data for:data for:

Σπ

Λπ

NK

NK

NKNK

Fitting ProceduresFitting ProceduresFitting ProceduresFitting Procedures

Fitting ProceduresFitting ProceduresFitting ProceduresFitting Procedures

ConstraintsConstraints

• 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(E0)+T’(E0)(E-E0) where

E is the CM energy of the data point in bin,E0 is the center energy in bin,

T(E0) is the complex T-matrix amplitude at CM energy E0,

T’(E0) is the “slope parameter” which is fixed at value from

unitarized solution

Results of Single-Energy Partial-Wave AnalysisResults of Single-Energy Partial-Wave Analysis

Results of Single-Energy Partial-Wave AnalysisResults of Single-Energy Partial-Wave Analysis

Results of Single-Energy Partial-Wave AnalysisResults of Single-Energy Partial-Wave Analysis

Results of Single-Energy Partial-Wave AnalysisResults of Single-Energy Partial-Wave Analysis

Results of Single-Energy Partial-Wave AnalysisResults of Single-Energy Partial-Wave Analysis

SummarySummary

What has been done:What has been done:

The available world database of The available world database of ddσσ/d/dΩΩ, total cross sections, , total cross sections, and polarization up to ~2 GeV, has been compiled, involving and polarization up to ~2 GeV, has been compiled, involving the reactions the reactions

Initialized with a set of unitarized partial-wave amplitudes, Initialized with a set of unitarized partial-wave amplitudes, after obtaining a reasonably smooth set of single-energy after obtaining a reasonably smooth set of single-energy solutions for the amplitudes, an energy-dependent fit was solutions for the amplitudes, an energy-dependent fit was carried out to ensure that the final results are consistent carried out to ensure that the final results are consistent with unitarity.with unitarity.

What remains to be done:What remains to be done:

Perform a unitarized fit based on our single-energy results Perform a unitarized fit based on our single-energy results Extract resonance parameters in a consistent manner for all Extract resonance parameters in a consistent manner for all channelschannels

ΣπΛπ NK,NKN,KNK

AcknowledgmentsAcknowledgments

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