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ppK - studied with a Chiral SU(3)-based K bar N potential

ppK - studied with a Chiral SU(3)-based K bar N potential. ´. A. Dote (KEK), W. Weise (TU Munich). T. Hyodo (TU Munich). Introduction Model - Simple Correlated Model (Revised) - Local K bar N potential based on Chiral SU(3) Result Summary and Future plan. Chiral ‘07

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ppK - studied with a Chiral SU(3)-based K bar N potential

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  1. ppK- studied with a Chiral SU(3)-based KbarN potential ´ A. Dote (KEK), W. Weise (TU Munich) T. Hyodo (TU Munich) • Introduction • Model- Simple Correlated Model (Revised) - • Local KbarN potential based on Chiral SU(3) • Result • Summary and Future plan Chiral ‘07 　’07.11.14 @ Osaka univ. Conventional Center

  2. １．Introduction Kbarnuclei = Exotic system !? I=0 KbarN potential … very attractive Highly dense state formed in a nucleus Interesting structures that we have never seen in normal nuclei etc… To know in more detail … ppK- = Prototye of Kbar nuclei • Studied with various methods, because it is a three-body system. • ATMS (Akaishi), Faddeev (Ikeda and Sato), Faddeev (Shevchenko and Gal), • Variational approach (Noda, Sasaki, Hiyama and Hirenzaki), Skyrme model (Nishikawa) … • Can be treated precisely • …a bare NN potential can be used, without help of G-matrix method. • Experiment done by FINUDA group • B.E. = 116 MeV, Γ = 67 MeV

  3. ２．Model Thanks, Akaishi-san! NN correlation function Single-particle motion Nucleon KN correlation function Kaon Model wave functionー Simple Correlated Model (Revised) ー … Nuclear partSpin=0, Isospin=1 … Nuclear partSpin=0, Isospin=0

  4. ２．Model Thanks, Akaishi-san! NN correlation function Single-particle motion Nucleon KN correlation function Kaon Model wave functionー Simple Correlated Model (Revised) ー Naïve “ppK-” … Nuclear partSpin=0, Isospin=1 … Nuclear partSpin=0, Isospin=0

  5. ２．Model Thanks, Akaishi-san! NN correlation function Single-particle motion Nucleon KN correlation function Kaon Model wave functionー Simple Correlated Model (Revised) ー … Nuclear partSpin=0, Isospin=1 … Nuclear partSpin=0, Isospin=0

  6. ２．Model Thanks, Akaishi-san! NN correlation function Single-particle motion Nucleon KN correlation function Kaon Model wave functionー Simple Correlated Model (Revised) ー … Nuclear partSpin=0, Isospin=1 … Nuclear partSpin=0, Isospin=0

  7. ２．Model NN potential…Av18 potential fitted with a few range Gaussians. 1E 1O

  8. ２．Model Influence of the improvement TN=1 + TN=0 TN=1 only Remark: NN potential is Tamagaki potential in Akaishi-san’s calculation. NN potential…Av18 potential Effective KbarN potential … Akaishi’s

  9. ３．Local KbarN potential based on Chiral SU(3) : CM energy of KbarN : Normalized Gaussian Go to Hyodo-san’s poster! Request for our KbarN potential 1. Reproduce the s-wave KbarN scattering amplitude calculated with Chiral unitary model To apply the structure study of ppK-, 2. Single channel (KbarN channel) but energy-dependent and complex 3. Local potential, r-space, Gaussian form

  10. ３．Local KbarN potential based on Chiral SU(3) Step 1 Chiral U. Chiral U. Single Ch. Coupled Ch. Vij Tij T = T11 • Step 1 • Eliminate other channels than KbarN channel π π K K K K K … K = + … + + … N N N N N N Σ Σ VSingle, I V11 Vj1 V1i • Exactly done in the framework of Chiral Unitary How to determine and the range parameter .

  11. ３．Local KbarN potential based on Chiral SU(3) Step 1 Step 2 Chiral U. Chiral U. Effective Single Ch. Coupled Ch. Single Ch. Vij Tij T = T11 • Step 2 • Using , construct simply as • Range parameter is determined so that • the I=0 resonance appears at the same place as that in the Chiral unitary • when we solve the Schroedinger equation with this potential. How to determine and the range parameter .

  12. ３．Local KbarN potential based on Chiral SU(3) Step 1 Step 2 Chiral U. Chiral U. Effective Single Ch. Coupled Ch. Single Ch. Vij Tij T = T11 Step 3 • Step 3 • Correct so as to reproduce the original scattering • amplitude (T-matrix) better especially far below the threshold. Corrected KbarN potential … “Corrected” KbarN potential without correction … “Uncorrected” How to determine and the range parameter .

  13. ３．Local KbarN potential based on Chiral SU(3) In Chiral unitary model, Resonance position in I=0KbarN channel 1420 MeV not 1405 MeV ! I=0KbarN scatteing amplitude “Uncorrected” “Corrected” Chiral Unitary Chiral Unitary 1420 The scattering amplitude far below threshold is overestimated if “Uncorrected” effective potential is used. (about twice) Chiral unitary;T.Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)

  14. ４．Result Kbar N N Kbar is bound by each nucleon with B(K)/2 binding energy. Notes on actual calculation • Hamiltonian Imaginary part of the effective KbarN potential is treated perturbatively. • We performed variational calculation. Since the KbarN potential is energy-dependent, we repeat the calculation until the self-consistency on the kaon energy is accomplished. • Binding energy of kaon : Hamiltonian of nuclear part • We have tried two prescriptions for . • We have tried “Corrected” and “Uncorrected” for four models: • “ORB” E. Oset, A. Ramos, and C. Bennhold, Phys. Lett. B527, 99 (2002) • “HNJH” T.Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003) • “BNW” B. Borasoy, R. Nissler, and W. Weise, Eur. Phys. J. A25, 79 (2005) • “BMN” B. Borasoy, U. G. Meissner, and R. Nissler, Phys. Rev. C74, 055201 (2006)

  15. ４．Result I=0 channel I=1 channel Notes on actual calculation • Hamiltonian Imaginary part of the effective KbarN potential is treated perturbatively. • We performed variational calculation. Since the KbarN potential is energy-dependent, we repeat the calculation until the self-consistency on the kaon energy is accomplished. • Binding energy of kaon : Hamiltonian of nuclear part • We have tried two prescriptions for . • We have tried “Corrected” and “Uncorrected” for four models: • “ORB” E. Oset, A. Ramos, and C. Bennhold, Phys. Lett. B527, 99 (2002) • “HNJH” T.Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003) • “BNW” B. Borasoy, R. Nissler, and W. Weise, Eur. Phys. J. A25, 79 (2005) • “BMN” B. Borasoy, U. G. Meissner, and R. Nissler, Phys. Rev. C74, 055201 (2006)

  16. ４．Result Total Binding Energy and Decay Width “Corrected”,

  17. ４．Result Total Binding Energy and Decay Width “Corrected”,

  18. ４．Result Total Binding Energy and Decay Width “Uncorrected”,

  19. ４．Result Total Binding Energy and Decay Width “Uncorrected”,

  20. ４．Result Total Binding Energy and Decay Width Total B. E. : 16 ～ 26 MeV Width : 37 ～ 63 MeV

  21. Structure of ppK- KbarN potential based on “HNJH” “Corrected”, Kbar N N

  22. Structure of ppK- KbarN potential based on “HNJH” “Corrected”, Kbar 2.00 fm N N 2.26 fm

  23. Structure of ppK- KbarN potential based on “HNJH” “Corrected”, Kbar N N TN=1 … 95.5 % TN=0 … 4.5 % T = 1/2 • NN distance = 2.26 fm KbarN distance = 2.00 fm

  24. Structure of ppK- KbarN potential based on “HNJH” “Corrected”, I=0 KbarN I=1 KbarN Kbar 1.83 fm 2.39 fm N N • NN distance = 2.26 fm KbarN distance = 2.00 fm • Mixture of TN=0 component = 4.5 %

  25. Structure of ppK- KbarN potential based on “HNJH” “Corrected”, “Λ(1405)” as I=0 KbarN calculated with this potential 1.87 fm I=0 KbarN I=1 KbarN Kbar 1.83 fm 2.39 fm Almost “Λ(1405)” N N • NN distance = 2.26 fm KbarN distance = 2.00 fm • Mixture of TN=0 component = 4.5 %

  26. Influence of P-wave KbarN potential For ap= 0.4～0.9 fm, VKN,P～ +3 MeV …small and repulsive • Estimate its contribution perturbatively. • Derived from “Full” scattering volume. The B(K) obtained with only the S-wave potential is very close to the position of Σ(1385) accidentally. (Slightly above it) 46 B(K) [MeV]

  27. ５．Summary and Future plan Summary • We have studied ppK- with a model that can treat the NN repulsive core directly. • Here, we have used Av18 potential as a bare NN potential. • We have constructed an effective s-wave KbarN potential which reproduces the scattering amplitude of • KbarN calculated in the framework of Chiral unitary model. • The present potential is energy-dependent, complex and local potential and has single Gaussian shape. • Exactly speaking, the system we are now considering is [NNKbar]T=1/2, Tz=1/2. • This system can contain TN (isospin of nuclear part) =0 component in addition to TN=1 component • which corresponds to just ppK-. Although the mixture of the TN=0 component is small, typically 5 %, • its contribution to the binding energy is rather large. • We have investigated the four KbarN potentials which are based on different Chiral unitary model • and tried two prescriptions for the relationship of √s and B(K). • For all cases, results are not so different: Total Binding energy = 16 ～ 26 MeV Width (S-wave) = 37 ～ 63 MeV … Very shallow binding • I=0 KbarN component in the ppK- seems almost genuine Λ(1405), investigating its size and • orbital angular momentum. This fact agrees with Akaishi-san’s picture. • We have estimated the influence of the p-wave KbarN potential, • derived from the “Full” scattering volume, perturbatively. • Since the system is shallowly bound, its contribution is small and repulsive. (VKN,P～ +3 MeV)

  28. ５．Summary and Future plan Future plan Understanding of the difference between our result and those obtained by other groups. • Comparison with Faddeev (Ikeda-san and Sato-san) study Total B. E. = 79 MeV, Decay width = 74 MeV Why is their result so different from ours, although their KbarN interaction is based on Chiral SU(3) theory similarly to our study? Is there a problem in the treatment of the energy dependence of two-body system (KbarN) in the three-body system (KbarNN)??? • Comparison with Faddeev (Shevchenko and Gal) study Total B. E. = 50 – 70 MeV, Decay width = ～100 MeV In their study, ppK- can be bound by about 40 MeV even only in the KbarNN channel, namely without coupling to the πΣN channel. Total B. E. = 48 MeV, Decay width = 61 MeV • Comparison with Akaishi-san’s study In the region of the sub-threshold, the absolute value of the KbarN scattering amplitude in addition to its behavior (energy-dependence) is definitely different from that derived from Chiral unitary model.

  29. ppK- calculated with various potentials Corrected Uncorrected √S = MN + mK - B(K) √S = MN + mK - B(K) / 2

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