New Multichannel Partial-Wave Analysis New Multichannel Partial-Wave Analysis Including Including ηηN and KN and KΛΛ Channels Channels

D. Mark ManleyD. Mark ManleyKent State UniversityKent State University

• IntroductionIntroduction• Physics MotivationPhysics Motivation• Single-Energy FitsSingle-Energy Fits• Global Energy-Dependent FitsGlobal Energy-Dependent Fits• ResultsResults• SummarySummary

PWA 2011 WorkshopPWA 2011 WorkshopThe George Washington UniversityThe George Washington UniversityWashington, DCWashington, DCMay 24, 2011May 24, 2011

IntroductionIntroduction

• Results will be presented for a new multichannel PWA of πN scattering in c.m. energy range 1080-2100 MeV.

• Includes new solutions for πN→ηN and πN→KΛ amplitudes determined from single-energy analyses.

• Global energy-dependent fits resulted in smooth amplitudes consistent with S-matrix unitarity.

• Resonance parameters obtained as results of global energy-dependent fits.

• Provides starting point for PWA of γN→ηN and γN→KΛ.

Physics MotivationPhysics Motivation

• Growing amount of high-quality data on γN→ηN and γN→KΛ is now becoming available from labs such as JLab, ELSA, MAMI, GRAAL, and LEPS.

• These data will lead to a clearer, more complete understanding of the N* spectrum.

• Analyses of πN→ηN and πN→KΛ complement the studies of η and kaon photoproduction.

Single-Energy FitsSingle-Energy Fits

• Previous PWAs of πN→ηN and πN→KΛ were usually simple energy-dependent and model-dependent parametrizations, such as T=TR+TB, which violated S-matrix unitarity.

• Energy-independent PWAs (i.e., single-energy fits) were normally not carried out due to insufficient data resulting in “noisy” solutions.

• We introduce an iterative method that combines the best of both techniques.

Single-Energy Fits (Cont’d)Single-Energy Fits (Cont’d)

• We began by assembling database of all available data for π-p→ηn and π-p→K0Λ.

• Data were analyzed in bins of width 30 MeV.

• Data for π-p→ηn spanned c.m. energies 1530±15 MeV to 2070±15 MeV and consisted of 443 dσ/dΩ measurements and 61 polarization measurements. About half the bins had no polarization data and four bins had no data at all.

• Data for π-p→K0Λ spanned c.m. energies 1648±15 MeV to 2068±15 MeV and consisted of 548 dσ/dΩ measurements and 260 polarization measurements. All bins had data and only one had no polarization data.

Single-Energy Fits (Cont’d)Single-Energy Fits (Cont’d)

• We began by fitting the lowest energy bins, which were dominated by the S11 wave. As we moved to higher energy bins, additional partial waves were added as needed until we reached the highest energy bin. The S11 wave for πN→ηN was held fixed below KΛ threshold at values from the energy-dependent GWU solution.

• The initial single-energy amplitudes provided excellent fits to the data but were noisy. These were then included in global energy-dependent fits with amplitudes for πN→πN, πN→ππN, and γN→πN.

• The global fits resulted in smooth amplitudes consistent with S-matrix unitarity, but the smoothed amplitudes no longer provided excellent agreement with dσ/dΩ and polarization data.

Single-Energy Fits (Cont’d)Single-Energy Fits (Cont’d)

• We next iterated the single-energy fits. Small amplitudes (with |T|<0.02) were held fixed at values from the global fits. This constraint resulted in somewhat smoother energy-dependent amplitudes.

• Some amplitudes were found to be small over the entire energy range. These were omitted in subsequent fits.

• For πN→ηN, the important partial waves were found to be S11, P11, P13, D15, F15, and G17.

• For πN→KΛ, the important partial waves were found to be S11, P11, P13, D13, and F15.

Single-Energy Fits (Cont’d)Single-Energy Fits (Cont’d)

• We iterated between the single-energy fits and the global energy-dependent fits. In subsequent iterations, we added additional constraints to the single-energy fits by holding some of the larger amplitudes fixed at the values from the global energy-dependent fits.

• Initially S11 and P11 amplitudes were held fixed, so the single-energy fits became more constrained and resulted in smoother values for the smaller amplitudes.

• In the next round, S11, P11, and F15 amplitudes were held fixed while the other amplitudes were varied.

• The procedure continued until only one or two amplitudes were free to vary in the final single-energy fits.

Single-Energy Fits (Cont’d)Single-Energy Fits (Cont’d)

• We resolved the global phase ambiguity in part by requiring the highest included partial wave to be real or almost real.

• For π-p→ηn, the global phase at low energies was determined from the “known” resonant S11 phase. At high energies, it was determined by choosing the almost real G17 amplitude to have the same phase as for πN elastic scattering.

• For π-p→K0Λ, the global phase was determined above 1.9 GeV by choosing the F15 wave to be real. Below 1.9 GeV, the global phase was determined by rotating amplitudes such that the P11 wave exhibited resonant behavior (Breit-Wigner phase) near c.m. energy of 1.7 GeV. This was consistent with the observed behavior of |T|2 for the P11 wave, which is independent of its phase.

Single-Single-Energy Fits: Energy Fits:

ddσσ/d/dΩΩ for for ππ--p→Kp→K00ΛΛ

Single-Single-Energy Fits: Energy Fits:

ddσσ/d/dΩΩ for for ππ--p→Kp→K00ΛΛ

Single-Single-Energy Fits: Energy Fits:

ddσσ/d/dΩΩ for for ππ--p→Kp→K00ΛΛ

σσ((ππ--p→Kp→K00ΛΛ))

Single-Single-Energy Fits: Energy Fits:

ddσσ/d/dΩΩ for for ππ--p→p→ηηnn

Single-Single-Energy Fits: Energy Fits:

ddσσ/d/dΩΩ for for ππ--p→p→ηηnn

Single-Single-Energy Fits: Energy Fits:

ddσσ/d/dΩΩ for for ππ--p→p→ηηnn

σσ((ππ--p→p→ηηn)n)

Global Energy-Dependent FitsGlobal Energy-Dependent Fits

• πN→πN amplitudes (SAID SP06 sol’n)

• γN→πN amlitudes (SAID FA07 sol’n)

• πN→ππN amplitudes (DMM et al., PRD 30, 904 (1984))

• πN→ηN (this work)

• πN→KΛ (this work)

• “KSU Model” was used to provide a unitary, multichannel parametrization of partial-wave amplitudes

• T.-S. H. Lee has shown this model can be derived starting from very general coupled-channel equations.*

__________________

*T.-S. H. Lee, in NSTAR 2005, Proceedings of the Workshop on the Physics of Excited Nucleons, p. 1 (World Scientific, 2006).

Global Energy-Dependent Fits (cont’d)Global Energy-Dependent Fits (cont’d)

• Fits determine BW masses and widths, pole positions, partial widths, decay amplitudes, and helicity amplitudes

• S11, P11, P13, D13, F15 – 10 channels

• D15 – 8 channels• P33, D33 – 7 channels• S31, F35 – 6 channels• D35 – 5 chann-Dels• P31, F37 – 4 channels• G17 – 3 channels• else – 2 channels

Parametrization of amplitudesParametrization of amplitudes

The KSU Model satisfies unitarity and time-reversal invariance. The total partial-wave S-matrix has the form

where the background where the background matrix matrix BB is unitary but not is unitary but not generally symmetric:generally symmetric:

The matrix The matrix RR is both is both unitary and symmetric. unitary and symmetric. It is a generalization of It is a generalization of the multichannel BW the multichannel BW form to include multiple form to include multiple resonances. It is con-resonances. It is con-structed from a K-structed from a K-matrix:matrix:

Parametrization of amplitudes (cont’d)Parametrization of amplitudes (cont’d)

For N resonances, K has the form

Elements of the matrices Elements of the matrices factorizable,factorizable,

where summing over all decay channels giveswhere summing over all decay channels gives

were assumed to bewere assumed to be

Parametrization of amplitudes (cont’d)Parametrization of amplitudes (cont’d)

For the special case of two resonances, we have

and the corresponding T-matrix has the formand the corresponding T-matrix has the form

where the coefficients can be calculated analytically. where the coefficients can be calculated analytically.

For further details, see For further details, see Baryon partial-wave analysisBaryon partial-wave analysis,,D.M. Manley, Int. J. Modern Phys. A D.M. Manley, Int. J. Modern Phys. A 1818, 441 (2003)., 441 (2003).

Results: SResults: S1111 and P and P1111 amplitudes amplitudes

Results: SResults: S1111 amplitudes amplitudes

Results: SResults: S1111 amplitudes amplitudes

Results: PResults: P1111 amplitudes amplitudes

Results: PResults: P1111 amplitudes amplitudes

Results: FResults: F15 15 and Gand G1717 amplitudes amplitudes

Results: FResults: F1515 amplitudes amplitudes

Results: GResults: G1717 amplitude amplitude

SummarySummary

• Single-energy amplitudes for πN→ηN and πN→KΛ have been determined by a constrained analysis up to a c.m. energy of almost 2.1 GeV.

• An energy-dependent solution consistent with S-matrix unitarity has been obtained, allowing resonance parameters to be determined in a consistent way for many channels including γN, πN, ηN, KΛ, and various quasi-two-body ππN channels.

• Predicted amplitudes for γN→πN and γN→KΛ provide a starting place for partial-wave analyses of those reactions.

AcknowledgmentsAcknowledgments

• Thanks to Igor Strakovsky and the other organizers of this meeting for inviting me to give this talk.

• Thanks to the GWU Data Analysis Center for providing files for πN→πN and γN→πN amplitudes and for πN→ηN data.

• This work could not have been done without the assistance of Manoj Shrestha.

• Funding for this work was provided in part by U.S. DOE Grant DE-FG02-01ER41194.

• Thank you for your attention!