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K - pp studied with Coupled-channel Complex Scaling method

K - pp studied with Coupled-channel Complex Scaling method. A. Doté (KEK ) and T. Inoue (Tsukuba univ .). Introduction Question --- Variational calculation and Faddeev --- Complex Scaling Method Result Summary and Future plan. Two-body K - p system … Λ(1405)

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K - pp studied with Coupled-channel Complex Scaling method

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  1. K-pp studied with Coupled-channel Complex Scaling method A. Doté(KEK) and T. Inoue (Tsukuba univ.) Introduction Question --- Variational calculation and Faddeev --- Complex Scaling Method Result Summary and Future plan Two-body K-p system … Λ(1405) studied with AY potential Not new calculation but confirmation of the past work Workshop on “Hadron and Nuclear Physics (HNP09)” ’09.11.19 @ Arata hall, Osaka univ., Ibaraki, Osaka, Japan

  2. I=0 KbarN potential … very attractive Deeply bound (Total B.E. ～100MeV) Highly dense state formed in a nucleus Interesting structures that we have never seen in normal nuclei… K- 1. Introduction Kbarnuclei(Nuclei with anti-koan) = Exotic system !? 3He K- ppn K- ppp Calculated with Antisymmetrized Molecular Dynamics method employing a phenomenological KbarN potential A. Doté, H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004) 3 x 3 fm2 0.0 0.15 0.0 2.0 0.0 2.0 [fm-3] To make the situation more clear … “K-pp”= Prototype of Kbarnuclei (KbarNN, Jp=1/2-, T=1/2)

  3. 1. Introduction A. Doté, T. Hyodo and W. Weise, Nucl. Phys. A804, 197 (2008) Phys. Rev. C79, 014003 (2009) Variational calculation of K-pp with a chiral SU(3)-based KbarN potential • Av18 NN potential … a realistic NN potential with strong repulsive core. • Effective KbarN potential based on Chiral SU(3) theory … reproduce the original KbarN scattering amplitude obtained with coupled channel chiral dynamics. Single channel, Energy dependent, Complex, Gaussian-shape potential I=0 KbarN resonance “Λ(1405)”appears at 1420 MeV, not 1405 MeV • Variational method I=0 KbarN resonance “Λ(1405)”appears at 1420 MeV, not 1405 MeV … Investigate various properties with the obtained wave function. × Four variants of chiral unitary models Total B. E. : 20 ± 3 MeV G（KbarN→pY） : 40 ～ 70 MeV NN distance = 2.2 fm KbarN distance= 2.0 fm ～　NN distance in normal nuclei Shallow binding and large decay width

  4. Recent results of calculation of K-pp and related experiments • Recent results of calculation of K-pp Width (KbarNN→πYN) [MeV] Doté, Hyodo, Weise [1] (Variational, Chiral SU(3)) Akaishi, Yamazaki [2] (Variational, Phenomenological) Shevchenko, Gal [3] (Faddeev, Phenomenological) - B.E. [MeV] Ikeda, Sato [4] (Faddeev, Chiral SU(3)) Exp. : DISTO [6] (Preliminary) Exp. : FNUDA [5] Using S-wave KbarN potential constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc. [1] PRC79, 014003 (2009) [2] PRC76, 045201 (2007) [3] PRC76, 044004 (2007) [4] PRC76, 035203 (2007) [5] PRL94, 212303 (2005) [6] arXiv:0810.5182 [nucl-ex]

  5. Recent results of calculation of K-pp and related experiments Width (KbarNN→πYN) [MeV] Doté, Hyodo, Weise [1] (Variational, Chiral SU(3)) Akaishi, Yamazaki [2] (Variational, Phenomenological) Shevchenko, Gal [3] (Faddeev, Phenomenological) - B.E. [MeV] Ikeda, Sato [4] (Faddeev, Chiral SU(3)) Exp. : DISTO [6] (Preliminary) Exp. : FNUDA [5] Using S-wave KbarN potential constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc. [1] PRC79, 014003 (2009) [2] PRC76, 045201 (2007) [3] PRC76, 044004 (2007) [4] PRC76, 035203 (2007) [5] PRL94, 212303 (2005) [6] arXiv:0810.5182 [nucl-ex]

  6. 2. Question --- Variational calc. and Faddeev --- ??? Discrepancy between Variational calc. and Faddeev calc. The KbarN potentials used in both calculations are constrained with Chiral SU(3) theory, but … Variational calculation (DHW) Faddeev calculation (IS) Total B. E. = 20±3 MeV, Decay width = 40～70 MeV Total B. E. = 60～95 MeV, Decay width = 45～80 MeV A. Doté, T. Hyodo and W. Weise, Phys. Rev. C79, 014003 (2009) Y. Ikeda, and T. Sato, Phys. Rev. C76, 035203 (2007) Why ? • Separable potential used in Faddeev calculation? • Non-relativistic (semi-relativistic) vs relativistic? • Energy dependence of two-body system (KbarN) in the three-body system (KbarNN)? • …???

  7. 2. Question --- Variational calc. and Faddeev --- A possible reason is πΣN thee-body dynamics Y. Ikeda and T. Sato, Phys. Rev. C79, 035201(2009) In the variational calculation (DHW), πΣ channel is eliminated and incorporated into the effective KbarN potential. π π K K K K K … K = + … + + … Two-body system N N N N N N Σ Σ V11 Vj1 V1i • This is correct for the two-body system. • The effective KbarN potential has energy dependence due to the elimination of πΣ channel. • It reproduces the original KbarN scattering amplitude calculated with KbarN-πΣ coupled channel model.

  8. 2. Question --- Variational calc. and Faddeev --- Apply the effective KbarN potential to the three-body system… Three-body system calculated with the effective KbarN potential Σ Σ π π K K K K K … K = + … + + … N N N N N N N N N N N N The energy of the intermediate πΣ state is fixed to the initial KbarN energy. Not true!

  9. 2. Question --- Variational calc. and Faddeev --- Apply the effective KbarN potential to the three-body system… Three-body system calculated with the effective KbarN potential Σ Σ π π N N N N N N N N K K K K K … K = + … + + … N N N N N N Due to the third nucleon, the intermediate πΣ energy can differ from the initial KbarN energy.

  10. 2. Question --- Variational calc. and Faddeev --- Apply the effective KbarN potential to the three-body system… Three-body system calculated with the effective KbarN potential π π K K K … K = + … N N N N Σ Σ conserved N N N N N N πΣN thee-body dynamics Due to the third nucleon, the intermediate πΣ energy can differ from the initial KbarN energy. Only the total energy of the three-body system should be conserved. The intermediate energy of the πΣ state can be variable, due to the third nucleon.

  11. 2. Question --- Variational calc. and Faddeev --- πΣN three-body dynamics is taken into account in the Faddeev calculation, because the πΣchannel is directly treated in their calculation. Dr. Ikeda‘s talk in KEK theory center workshop “Nuclear and Hadron physics” (KEK, 11-13 Aug. ‘09)

  12. 3. Complex Scaling method Direct treatment of the “πΣ” degree of freedom Coupled channel calculation A bound state for KbarNN channel, Kbar + N + N π + Σ + N but a resonant state for πΣN channel “Kbar N N” Resonant state can’t be treated with a variational method. Here, we employ “Complex Scaling method” to deal with resonant state. • The actual calculation is done in multi channels such as KbarN and πΣ. • The obtained state is possibly to appear below KbarNN threshold, but above πΣN threshold.

  13. 3. Complex Scaling method Advised by Dr. Myo (RCNP) Complex rotation of coordinate The energy of bound and resonant states is independent of scaling angle θ. (ABC theorem) Wave function of a resonant state (two-body system) Negative! When, the wave function of a resonant state changes to a dumping function. Boundary condition is the same as that for a bound state. The resonant state can be obtained by diagonalizingwith Gaussian basis, quite in the same way as calculating bound state.

  14. 4. Result • Two-body system: KbarN-πΣ coupled system with s-wave and isospin zero state … Λ(1405) Here, we consider just two-channel problem; KbarN and πΣ channels Schrödinger equation to be solved Employ Akaishi-Yamazaki potential Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 Wave function expanded with Gaussian base : complex parameters to be determined Complex-rotate , then diagonalize with Gaussian base.

  15. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] 2q -G / 2 q= 0 deg.

  16. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] 2q -G / 2 q= 5 deg.

  17. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] 2q -G / 2 q=10 deg.

  18. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 2q q=15 deg.

  19. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 2q q=20 deg.

  20. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 q=25 deg.

  21. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 q=30 deg. 2q

  22. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 q=35 deg. 2q

  23. 4. Result qtrajectory • # Gauss base (n) = 30 • Max range (b) = 10 [fm] E [MeV] -G / 2 q=40 deg. 2q

  24. 4. Result qtrajectory Resonance! (E, Γ/2) = (75.8, 20.0) pS KbarN E [MeV] • Measured from KbarN thr., • B. E. (KbarN) = 28.2 MeV • Γ = 40.0 MeV • … L(1405) ! KbarN continuum -G / 2 pScontinuum 2q q=30 deg.

  25. 4. Result Stability of the resonance for other parameters Analysis ofθtrajectory with • # Gauss base (n) = 30 • Max range (b) = 10 [fm] • Resonance found at • B. E. (KbarN) = 28.2 MeV • Γ = 40.0 MeV b trajectory • # Gauss base (n) = 30 • Rotation angle (q) = 30 [deg] n trajectory • Max range (b) = 10 [fm] • Rotation angle (q) = 30 [deg] b = 50 [fm]

  26. 5. Summary and Future plan Complex Scaling Method applying to Λ(1405) • Solve the Coupled Channel problem = KbarN + πΣ • In case of AY potential, • Resonance found at • B. E. (KbarN) = 28.2 MeV • Γ = 40.0 MeV Comment • No other resonant state (no interaction between πand Σ) • In case of only VKN-KN, a bound state appears at B. E. = 3.2 MeV. VKN-KN • Coupling with πΣcontinuum state increases • drastically the binding energy. • Also, it gives the decay width. VKN-πΣ KbarN thr.

  27. 5. Summary and Future plan Future plan Keep doing calculations! and thinking… 1. Calculate two-body system … KbarN-πΣ system corresponding to Λ(1405) • For a test calculation, a phenomenological KbarN potential • (AY potential, energy-independent) will be employed. Investigate the property of the obtained wave function Done Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 • See what happen if we use a chiral SU(3)-based KbarN potential • (HW potential, energy-dependent). T. Hyodo and W. Weise, PRC77, 035204(2008) 2. Calculate three-body system … KbarNN-πΣN system corresponding to “K-pp”

  28. J-PARC will give us lots of interesting data! E15: A search for deeply bound kaonic nuclear states by 3He(inflightK-, n) reaction --- Spokespersons: M. Iwasaki (RIKEN), T. Nagae (Kyoto) E17: Precision spectroscopy of kaonic3He atom 3d→2pX-rays --- Spokespersons: R. Hayano (Tokyo), H. Outa (Riken)

  29. Thank you very much! Acknowledgement: Dr. Myo (RCNP) taught me CSM and supplied a subroutine for diagonalization of a complex matrix.

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