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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). K 中間子原子核. T. Hyodo (TU Munich). Introduction Model - Simple Correlated Model (Revised) - Local K bar N potential based on Chiral SU(3) Result Summary and Comment. 「原子核・ハドロン物理：横断研究会」

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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) K中間子原子核 T. Hyodo (TU Munich) • Introduction • Model- Simple Correlated Model (Revised) - • Local KbarN potential based on Chiral SU(3) • Result • Summary and Comment 「原子核・ハドロン物理：横断研究会」 　’07.11.19 @ KEK 4号館1階セミナーホール

  2. １．Introduction I=0 KbarN potential … very attractive Highly dense state formed in a nucleus Interesting structures that we have never seen in normal nuclei… K- Akaishi, Yamazaki ATMS B.E. = 47MeV, Γ= 61MeV PRC76, 045201(2007) Ikeda, Sato Faddeev B.E. = 79MeV, Γ= 74MeV PRC76, 035203(2007) Shevchenko, Gal Faddeev B.E. = 50～70MeV, Γ=～100MeV PRC76, 044004(2007) Arai, Yasui, Oka (Λ* nuclei model), Nishikawa, Kondo (Skyrme model), Noda, Yamagata, Sasaki, Hiyama, Hirenzaki (Variational method) … FINUDA experiment B.E. = 116MeV, Γ= 67MeV PRL94, 212303(2005) Kbarnuclei = Exotic system !? To know in more detail … ppK- = Prototye of Kbar nuclei Studied with various methods, because it is a three-body system:

  3. １．Introduction Super Strong Nuclear Force Y. Akaishi and T. Yamazaki, PRC76, 045201(2007)

  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 … 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 Naïve “ppK-” … 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 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

  8. 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

  9. ２．Model NN potential…Av18 potential fitted with a few range Gaussians. 1E 1O

  10. 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

  11. ３．Local KbarN potential based on Chiral SU(3) : CM energy of KbarN : Normalized Gaussian Hyodo-san’s work 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

  12. 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 .

  13. 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 .

  14. 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 .

  15. 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)

  16. ４．Result • Hamiltonian is treated perturbatively. • Binding energy of kaon : Hamiltonian of nuclear part Notes on actual calculation • 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.

  17. ４．Result • Hamiltonian is treated perturbatively. Kbar • Binding energy of kaon N N : Hamiltonian of nuclear part • We have tried two prescriptions for . Kbar is bound by each nucleon with B(K)/2 binding energy. Notes on actual calculation • 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. • 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)

  18. ４．Result is treated perturbatively. I=0 channel I=1 channel Notes on actual calculation • Hamiltonian • 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 . 1420 • 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)

  19. ４．Result is treated perturbatively. Notes on actual calculation • Hamiltonian • 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)

  20. ４．Result Total Binding Energy and Decay Width “Corrected”,

  21. ４．Result Total Binding Energy and Decay Width “Corrected”,

  22. ４．Result Total Binding Energy and Decay Width “Uncorrected”,

  23. ４．Result Total Binding Energy and Decay Width “Uncorrected”,

  24. ４．Result Total Binding Energy and Decay Width Total B. E. : 21 ± 5 MeV Width : 50 ± 13 MeV

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

  26. Structure of ppK- KbarN potential based on “HNJH” “Corrected”, Cf) NN distance in normal nuclei ～ 2 fm Size of deuteron ～ 4 fm Kbar 2.00 fm N N 2.26 fm

  27. 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

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

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

  30. Structure of ppK- KbarN potential based on “HNJH” “Corrected”, Kbar I=0 KbarN I=1 KbarN 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 %

  31. 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 1.83 fm 2.39 fm Kbar Almost “Λ(1405)” N N • NN distance = 2.26 fm KbarN distance = 2.00 fm • Mixture of TN=0 component = 4.5 %

  32. 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]

  33. ５．Summary NN potential = Av18 potential … Stongly repulsive core KbarN potential = Local potential based on Chiral unitary model … Scattering amplitude is reproduced. Single Gaussian form Strong energy dependence Simple correlated model (Revised) … Respect Short-range correlation. Contain TN=0 component as well as TN=1 component. Four Chiral unitary models and Two prescriptions on B(K) Total Binding energy = 21 ± 5 MeV Width (S-wave) = 50 ± 13 MeV … Very shallow binding ppK- is studied with + • 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)

  34. ５．Summary Smaller than deuteron, comparable to NN in normal nuclei. I=0 KbarN in ppK- is almost “Λ(1405)” ! … This picture doesn’t contradict with Akaishi-san’s one. Structure of ppK- • NN distance = ～2.2 fm • KbarN distance = ～2.0 fm • Mixture of TN=0 component = 4.5 % Genuine ppK- → TN=1 component • KbarN in [ppK-]T=1/2 “Λ(1405)” Size = 1.83 fm, < l2 >=0 I=0 : Distance = 1.83 fm, < l2 >=0.3 I=1 : Distance = 2.39 fm, < l2 >=1.9 • P-wave KbarN potential : small and repulsive (～ +3 MeV)

  35. ５．Comment No predictive power for KbarN subthreshold, even in Chiral unitary model… We need experimental data for this region. Kaonic hydrogen atom KpX, DEAR ??? Scattering data KbarN No data KbarN threshold Σπ mass spectrum Σ+π- and Σ-π+ for I=0 Σπ No data ofΣ0π0 Σπ threshold

  36. ５．Comment D D 深いっすよねー I’m not a bat! I want only to try various possibilities. Only the experiment can give an answer, I think. So, we expect J-PARC!! It should be shallowly bound.

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