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The “K - pp” system investigated with a fully coupled-channel Complex Scaling Method

This study investigates the K-pp system using a fully coupled-channel Complex Scaling Method. The energy dependence of the Chiral SU(3)-based potential is considered and the relativistic effects are analyzed. Collaborators include Takashi Inoue and Takayuki Myo.

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The “K - pp” system investigated with a fully coupled-channel Complex Scaling Method

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  1. The “K-pp”system investigated with a fully coupled-channel Complex Scaling Method AkinobuDoté (KEK Theory Center, IPNS / J-PARC branch) • Introduction • Formalism • Fully coupled-channel Complex Scaling Method for “K-pp” • How to treat the energy-dependence of Chiral SU(3)-based potential • Result • Relativistic effect • Summary and future prospects • Collaborators: • Takashi Inoue (Nihon university) • Takayuki Myo (Osaka Institute of Technology) The 15th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU 2019) June 7, 2019 @ Carnegie Mellon University in Pittsburgh, Pennsylvania, USA

  2. The “K-pp” system investigated with a coupled-channel Complex Scaling Method Clock Tower, Kyoto University

  3. The “K-pp” system investigated with a coupled-channel Complex Scaling Method The “K-pp” system investigated with a fully coupled-channel Complex Scaling Method

  4. 1. Introduction

  5. Λ(1405) ... KbarN quasi-bound state = KbarN two-body system Proton → Strong KbarN attraction T. Hyodo, D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012) K- K- K- pp = the prototype of kaonicnuclei ... Bridge from Λ(1405) to general kaonic nuclei P Kaonic nuclei = Nuclear many-body system with antikaons P Doorway to dense matter? → Chiral symmetry restoration, Neutron star physics, ... A. Cieply, E. Friedman, A. Gal, D. Gazda, J. Mares, PRC 84, 045206 (2011) T. Muto, T. Maruyama, T. Tatsumi, JPS Conf. Proc. 17, 102003 (2017)

  6. Status of “K-pp” studies (as of 2016) J-PARC E27 Theory: K-pp should be bound! DISTO Faddeev-AGS Phen. pot. (E-indep.) J-PARC E15 (1st run) Chiral pot. (E-dep.) → Small B. E. FINUDA Variational (Gauss) Phen. pot. (E-indep.) Phenomenological pot. (E-indep.) → Large B. E. Variational (Gauss) Chial. pot. (E-dep.) Variational (H.H.) Faddeev-AGS Chiral pot. (E-dep.) Summarized in A. Gal, E. V. Hungerford, D. J. Millener, Rev. Mod. Phys. 88, 035004 (2016).

  7. Many theoretical studies of “K-pp” suggest ... “K-pp” (Jπ=0-, T=1/2) • Variational / Phen. pot. Akaishi, Yamazaki, PRC76, 045201(2007) • Variational / Chiral pot. Doté, Hyodo, Weise, PRC79, 014003(2009) • Variational / Chiral pot.Barnea, Gal, Liverts, PLB712, 132(2012) • Faddeev-AGS / Phen. pot. Shevchenko, Gal, Mares, PRC76, 044004(2007) • Faddeev-AGS / Chiral pot. Ikeda, Kamano, Sato, PTP124, 533(2010) Resonant state of KbarNN-πΣN-πΛN coupled channel three-body system Resonance & Channel coupling ⇒ “Fully coupled-channel Complex Scaling Method” A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)

  8. 2. Formalism • Fully coupled-channel Complex Scaling Method for “K-pp” • How to treat the energy-dependence of Chiral SU(3)-based potential MENU 2016 πYN channels: indirectly treated by means of Feshbach projection A. D., T. Inoue, T. Myo, PTEP 2015, 043D02 “K-pp”= KbarNN– πΣN – πΛN (Jπ = 0-, T=1/2)

  9. 2. Formalism • Fully coupled-channel Complex Scaling Method for “K-pp” • How to treat the energy-dependence of Chiral SU(3)-based potential MENU 2016 πYN channels: indirectly treated by means of Feshbach projection A. D., T. Inoue, T. Myo, PTEP 2015, 043D02 MENU 2019 πYN channels: directly treated “K-pp”= KbarNN– πΣN – πΛN (Jπ = 0-, T=1/2)

  10. Wave function ...Treat all channels explicitly! • Baryon-Baryon are antisymmetrized on space, spin and isospin as well as label (flavor).Glöckle, Miyagawa, Few-body Systems 30, 241 (2001) • Spatial part = Correlated Gaussian function • including 3 types of Jacobi coordinates • projected onto a parity eigenstate of B1B2, M3 B1 x2(3) x1(3) B2

  11. Complex Scaling Method … Find resonance poles on complex energy plane! S. Aoyama, T. Myo, K. Kato, K. Ikeda, PTP116, 1 (2006) T. Myo, Y. Kikuchi, H. Masui, K. Kato, PPNP79, 1 (2014)

  12. Full ccCSM with a phen. potential NN: Av18 pot. KbarN-πY: Akaishi-Yamazaki pot. (Phen., E-indep.) Λ* pole BKN = 28 MeV, ΓπΣ /2 = 20 MeV The “K-pp” pole for AY-potential case: BKNN = 51 MeV, ΓπYN /2 = 16 MeV KbarNN cont. Λ*N cont. Scaling angle θ=30° Dimension = 6400 πΛN cont. πΣN cont. A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)

  13. A. Dote, T. Inoue, T. Myo, PLB 784, 405 (2018) • 2. Formalism • Fully coupled-channel Complex Scaling Method for “K-pp” • How to treat the energy-dependence of Chiral SU(3)-based potential

  14. Hamiltonian 3 1 Meson Baryon 2 Baryon • Kinematics = Non-relativistic • NN potential = Av18 potential • KbarN-πY potential ... Chiral SU(3)-based potential R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, PRC 51, 38 (1995) • Theoretical and energy-dependent potential • Gaussian form potential • Constrained by the latest KbarN scattering length • A. Dote, T. Inoue, T. Myo, NPA 912, 66 (2013) + SIDDHARTA K-p data M. Bazzi et al., NPA 881, 88 (2012) • Coupled-channel chiral dynamics • Y. Ikeda, T. Hyodo, W. Weise, NPA 881, 98 (2012) • Ignore YN and πN potentials

  15. How to deal with E-dep. potential? • How to treat energy-dependent potentials in many-body system? • In addition, in coupled-channels cases??? “Self-consistency for Meson-Baryon energy” ... generalize a prescription for single-channel cases (Dote, Hyodo, Weise, PRC79, 014003 (2009))

  16. Self-consistency for KbarN energy • KbarNN single-channel case Estimation of the two-body energy in the three-body system A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009) 1. Kaon’s binding energy: : Hamiltonian of 2N : Hamiltonian of 2N+Kbar An interacting KbarN pair carries 2. Calculate the KbarN energy in two ways: : Field picture ... 100% of B(K) ... 50% of B(K) : Particle picture

  17. Self-consistency for MB energy • KbarNN-πＹＮcoupled-channel case Consider the coupled-channel Hamiltonian and mass operators, Calculate their expectation values with the obtained wave function. Using these values, we estimate the averaged MB energy. Estimation of the two-body energy in the three-body system A. D., T. Inoue, T. Myo, PLB784, 405 (2018) 1. Meson’s binding energy: 2. Calculate the MB energy in two ways: : Field picture : Particle picture

  18. A. Dote, T. Inoue, T. Myo, PLB 784, 405 (2018) 3. Result a chiral SU(3)-based potential

  19. Binding energy and decay width Remark For each fπ value, range parameters in the potential are tuned to reproduce the KbarN scattering length. -BK-pp [MeV] fπ = 120 MeV • Field picture: BK-pp = 14 － 28 MeVΓπYN /2 = 8 － 15 MeV • Particle picture: BK-pp = 21 － 50 MeVΓπYN /2 = 13 － 19 MeV -ΓπYN /2 [MeV] 90 90 120 SIDDHARTA constraint

  20. NN: Av18 pot. KbarN-πY: Chiral pot. w/ SIDDARTA const. (fπ=110 MeV / Field pict.) Property of the “K-pp” pole Norm -0.128 + i0.041 -0.002 + i0.004 1.131- i0.045 πΛ (I=0)--- (I=1) -0.002 + i0.004 πΣ (I=0) -0.125 + i0.033 (I=1) -0.004 + i0.008 KbarN (I=0) 0.826 - i0.034 (I=1) 0.304 - i0.011 BB distance [fm] 1.67+ i0.30 2.14- i0.16 1.22+ i0.12 MB distance [fm] (I=0) --- (I=1) 0.63 + i0.51 (I=0) 0.42 + i1.08 (I=1) 0.49 + i1.04 (I=0) 1.70 - i0.21 (I=1) 2.27 - i0.22

  21. NN: Av18 pot. KbarN-πY: Chiral pot. w/ SIDDARTA const. (fπ=110 MeV / Field pict.) Property of the “K-pp” pole Norm -0.128 + i0.041 -0.002 + i0.004 1.131- i0.045 πΛ (I=0)--- (I=1) -0.002 + i0.004 πΣ (I=0) -0.125 + i0.033 (I=1) -0.004 + i0.008 KbarN (I=0) 0.826 - i0.034 (I=1) 0.304 - i0.011 BB distance [fm] ⇒Half of deuteron size (~4 fm) 1.67+ i0.30 2.14- i0.16 1.22+ i0.12 MB distance [fm] (I=0) --- (I=1) 0.63 + i0.51 (I=0) 0.42 + i1.08 (I=1) 0.49 + i1.04 (I=0) 1.70 - i0.21 (I=1) 2.27 - i0.22

  22. NN: Av18 pot. KbarN-πY: Chiral pot. w/ SIDDARTA const. (fπ=110 MeV / Field pict.) Property of the “K-pp” pole Norm -0.128 + i0.041 -0.002 + i0.004 1.131- i0.045 πΛ (I=0)--- (I=1) -0.002 + i0.004 πΣ (I=0) -0.125 + i0.033 (I=1) -0.004 + i0.008 KbarN (I=0) 0.826 - i0.034 (I=1) 0.304 - i0.011 BB distance [fm] 1.67+ i0.30 2.14- i0.16 1.22+ i0.12 cf) Λ(1405) ~ KbarN (I=0) ... 1.46 fm MB distance [fm] (I=0) --- (I=1) 0.63 + i0.51 (I=0) 0.42 + i1.08 (I=1) 0.49 + i1.04 (I=0) 1.70 - i0.21 (I=1) 2.27 - i0.22

  23. Examination on Energy Dependence • Fix the meson-baryon energy in the chiral potential to the KbarN-threshold energy • With SIDDHARTA constraint ※ fπ = 110MeV Λ* pole I=0 KbarN scattering amplitude • E-dep. case (Solid line): BKbar-N = 8.1 MeVΓ/2 = 13.3 MeV [fm] • E-fixed case (Dashed line): BKbar-N = 5.7 MeVΓ/2 = 15.5 MeV √s [MeV] K-pp in E-fixed case (fπ= 90 － 120 MeV): BK-pp = 5 － 23 MeV ΓπYN /2 = 26 － 39 MeV Small binding energy less than 25 MeV, similar to the Field-picture case

  24. Current result of “K-pp” J-PARC E15-2nd PLB 789, 620 (2019) SGM Non-mesonicdecay mode (KbarNN→YN)? AY DHW IKS BGL Full ccCSM w/ AY pot. (Coupled) Particle Field Full ccCSM w/ Chiral SU(3) pot.(SIDDHARTA)

  25. 4. Relativistic effect Small mass!  Is Non-rela. treatment o.k.?

  26. Semi-relativistic treatment of mesons (π and Kbar) • Meson ... Semi-relativistic kinematics (SR) • Baryon ... Non-relativistic kinematics (NR) • KbarN-πY potential is reconstructed in SR kinematics. Pion appears in decay channels.  Change of kinematics may affect the decay width??? Case: SIDDHARTA / Field picture / fπ=110 MeV • Non-rela.(BK-pp, ΓπYN/2) = (17.5, 10.0) [MeV] • Semi-rela.(Meson)(BK-pp, ΓπYN/2) = (30.2, 29.3) Preliminary Decay width increases.

  27. 5. Summary and Future prospects

  28. 5. Summary K-pp = a prototype of kaonic nuclei Binding energy and mesonic decay width of “K-pp” (Jπ=0-, T=1/2) : • Chiral SU(3)-based KbarN potential constrained with the latest KbarNdata (SIDDAHRTA) K-pp is completely treated as a resonance of KbarNN-πΣN-πΛN coupled-channel system byFully coupled-channel Complex Scaling Method. (BK-pp, ΓπYN /2) = (14--28, 8--15) MeV (Field picture) (21--50, 13--19) MeV (Particle picture) Self-consistency for meson-baryon energy is considered. • Phenomenological KbarN potential (Akaishi-Yamazaki potential; Energy-independent) (BK-pp, ΓπYN /2) = (51, 16) MeV • Examined fixing the meson-baryon energy of the chiral potential Under the SIDDHARTA constraint, the K-pp is shallowly bound. ( BK-pp < 25 MeV) • Examined Semi-relativistic kinematics for mesons (Preliminary) Decay width (ΓπYN) seems to become larger, compared with Non-relativistic case.

  29. 5. Future prospects • Semi-relativistic treatment(Undergoing) • Examine other KbarN potentials… A sophisticated chiral SU(3)-based potential (K. Miyahara, T. Hyodo and W. Weise, PRC 98, 025201 (2018)) • Non-mesonic decay (KbarNN→YN)... Sizable contribution to the decay width (M. Bayar, E. Oset, Phys. Rev. C 88, 044003 (2013)) • Reaction spectrum... Direct comparison with experimental data.ex) T. Sekihara, E. Oset and A. Ramos, PTEP 2016, 123D03 (2016); arXiv:1903.10773. Thank you very much! • References: • A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017) • A. Dote, T. Inoue, T. Myo, PLB 784, 405 (2018)

  30. Backup slides

  31. Λ(1405) ... KbarN quasi-bound state = KbarN two-body system Proton → Strong KbarN attraction T. Hyodo, D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012) K- K- K- pp = the prototype of kaonicnuclei ... bridge from Λ(1405) to general kaonic nuclei P P Kaonic nuclei = Nuclear many-body system with antikaons Doorway to dense matter? → Chiral symmetry restoration, Neutron star physics

  32. 5. Future prospects • Semi-relativistic treatment(Undergoing) ... Pion mass is small. Influence to decay width? • Examine other KbarN potential … A sophisticated version of a chiral SU(3)-based KbarN-πY local potential K. Miyahara, T. Hyodo and W. Weise, PRC 98, 025201 (2018) • Non-mesonic decay (KbarNN→YN) • Reaction spectrum ... Direct comparison with experimental data. It can be calculated using the Green function obtained with ccCSM. (“Morimatsu-YazakiGreen’s function method”) • ...

  33. Thank you very much! • References: • A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017) • A. Dote, T. Inoue, T. Myo, PLB 784, 405 (2018)

  34. Property of the “K-pp” pole NN: Av18 pot. KbarN-πY: Akaishi-Yamazaki pot. (Pheno., E-indep.) Norm -0.002 + i 0.276 -0.002 + i 0.010 1.004 - i 0.286 πΛ (I=0)--- (I=1) -0.002 + i 0.010 πΣ (I=0) -0.009 + i 0.261 (I=1) 0.007 + i 0.015 KbarN (I=0) 0.726 - i 0.213 (I=1) 0.278 - i 0.073 BB distance [fm] 1.50+ i 0.48 1.86 + i 0.14 1.10+ i 0.23 MB distance [fm] (I=0) --- (I=1) 0.74 + i 0.74 (I=0) 0.52 + i 1.31 (I=1) 0.80 + i 1.22 (I=0) 1.40 - i 0.01 (I=1) 1.95 + i 0.14

  35. Property of the “K-pp” pole NN: Av18 pot. KbarN-πY: Akaishi-Yamazaki pot. (Pheno., E-indep.) Norm -0.002 + i 0.276 -0.002 + i 0.010 1.004 - i 0.286 πΛ (I=0)--- (I=1) -0.002 + i 0.010 πΣ (I=0) -0.009 + i 0.261 (I=1) 0.007 + i 0.015 KbarN (I=0) 0.726 - i 0.213 (I=1) 0.278 - i 0.073 BB distance [fm] NN distance = 1.9 fm ⇒1.6 ρ0 1.50+ i 0.48 1.86 + i 0.14 1.10+ i 0.23 MB distance [fm] (I=0) --- (I=1) 0.74 + i 0.74 (I=0) 0.52 + i 1.31 (I=1) 0.80 + i 1.22 (I=0) 1.40 - i 0.01 (I=1) 1.95 + i 0.14

  36. Property of the “K-pp” pole NN: Av18 pot. KbarN-πY: Akaishi-Yamazaki pot. (Pheno., E-indep.) Norm -0.002 + i 0.276 -0.002 + i 0.010 1.004 - i 0.286 πΛ (I=0)--- (I=1) -0.002 + i 0.010 πΣ (I=0) -0.009 + i 0.261 (I=1) 0.007 + i 0.015 KbarN (I=0) 0.726 - i 0.213 (I=1) 0.278 - i 0.073 BB distance [fm] 1.50+ i 0.48 1.86 + i 0.14 1.10+ i 0.23 cf) Λ(1405) ~ KbarN (I=0) ... 1.25 fm MB distance [fm] (I=0) --- (I=1) 0.74 + i 0.74 (I=0) 0.52 + i 1.31 (I=1) 0.80 + i 1.22 (I=0) 1.40 - i 0.01 (I=1) 1.95 + i 0.14

  37. Experimental search for K-pp • Deeply bound region (near πΣN threshold = 103 MeV below KbarNN threshold)FINUDA (2005), DISTO (2010), J-PARC E27 (2013) • Shallowly bound region (near KbarNN threshold)J-PARC E15 1st run (2013) • No signal in bound regionLEPS/SPring8 (2015), HADES (2015) and more ... Prof. Iwasaki’s talk (Morning on Friday) Clear evidence of K-pp bound state J-PARC E15 (2nd run): Exclusive exp. 3He(K-, Λp)nmissing

  38. Theoretical studies of “K-pp” Many theoretical studies of K-pp! and more ...

  39. Theoretical studies of “K-pp” • K-pp should be bound. BK-pp < 100 MeVResonance between πΣN and KbarNN thresholds. J-PARC E27 DISTO Faddeev-AGS Pheno. pot. (E-indep.) J-PARC E15 Chiral pot. (E-dep.) → Small B. E. FINUDA Variational (Gauss) Pheno. pot. (E-indep.) Pheno. pot. (E-indep.) → Large B. E. Variational (Gauss) Chial. pot. (E-dep.) Faddeev-AGS Chiral pot. (E-dep.)

  40. Just diagonalize the complex-scaled Hamiltonian matrix! KbarNN - KbarNN KbarNN - πΛN KbarNN - πΣN θ Hij πΣN - πΣN πΣN - πΛN ＊ πΛN - πΛN ＊ ＊ No channel elimination!!

  41. Hamiltonian 3 1 Meson Baryon 2 Baryon ... symmetric for baryon’s site (i=1,2) Glöckle, Miyagawa, Few-body Systems 30, 241 (2001) • Kinematics = Non-relativistic • NN potential = Av18 potential • KbarN-πY potential = Chiral SU(3)-based potential R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, PRC 51, 38 (1995) • A. Dote, T. Inoue, T. Myo, NPA 912, 66 (2013) Theoretical energy-dependentpotential : Gaussian form • Ignore YN and πN potentials

  42. 2. Formalism • Fully coupled-channel Complex Scaling Method for “K-pp” • Chiral SU(3)-based potential • Self-consistency for energy-dependent potential in coupled-channel case

  43. Chiral SU(3)-based KbarN potential • Anti-kaon, Pion = Nambu-Goldstone boson ... governed by chiral dynamics • Coupled-channel chiral dynamics(Chiral Unitary model)N. Kaiser, P.B. Siegel, W. Weise, NPA 594 (1995) 325E. Oset, A. Ramos, NPA 635 (1998) 99 • Weinberg-Tomozawaterm of effective chiral Lagrangian • Based on Chiral SU(3) theory • → Energy dependence • Constrained by KbarN scattering length • Old data:aKN(I=0) = -1.70 + i0.67 fm, aKN(I=1) = 0.37 + i0.60 fmA. D. Martin, NPB179, 33(1979) • SIDDHARTA K-p data with a coupled-channel chiral dynamics:aK-p = -0.65 + i0.81 fm, aK-n = 0.57 + i0.72 fm M. Bazzi et al., NPA 881, 88 (2012) Y. Ikeda, T. Hyodo, W. Weise, NPA 881, 98 (2012)

  44. Self-consistent solutions: MB energy assumed in the potential = MB energy estimated with the obtained wave function Result Martin constraint -BK-pp [MeV] Remark For each fπ value, range parameters in the potential are tuned to reproduce the KbarN scattering length. fπ = 120 MeV • Field picture: BK-pp = 19 － 37 MeVΓπYN/2 = 8 － 14 MeV • Particle picture: BK-pp = 29 － 59 MeVΓπYN/2 = 11 － 14 MeV 90 120 90 -ΓπYN/2 [MeV]

  45. 4. Discussion ! ...??

  46. 1. Dense matter or not? • Pheno. potential • (E-indep.) • Y. Akaishi and T. Yamazaki, PRC 65, 044005 (2002) • Martin constraint • Chiral SU(3) potential • (E-dep.) • A. Dote, T. Inoue and T. Myo, NPA 912, 66 (2013) • SIDDHARTA constraint K- P Dense matter B. E. (Λ*) = 28 MeV ⇒ Λ* = Λ(1405) B. E. (Λ*) ~ 15 MeV ⇒ Λ* = Λ(1420) Normal matter B. E. (“K-pp”) = 14-28 MeV NN distance = 2.1 fm (Field picture) P P B. E. (“K-pp”) = 51 MeV NN distance = 1.9 fm K- ⇒ ~ 1.6 ρ0 ⇒ ~ ρ0 (=0.17 fm-3)

  47. 2. “Comparison”with experimental results E15 2nd arXiv: 1805.12275 FINUDA DISTO J-PARC E27 SGM • The binding energy of K-pp observed in E15 2ndis difficult to be reproduced with Field picture.(Particle picture seems possible...) AY DHW IKS BGL Full ccCSM w/ AY pot. (Coupled) Particle Field Full ccCSM w/ Chiral SU(3) pot.(SIDDHARTA) • The observed decay widthis much larger. ⇒ Non-mesonic decay mode (KbarNN→YN)?Remark) Width obtained in calculations = Mesonic decay width (KbarNN→πYN)

  48. Full ccCSM with Semi-Relativistic Hamitonian , where -BK-pp [MeV] Preliminary ! Semi-Relativistic Hamiltonian... Kinetic energy is given in relativistic form. In semi-relativistic calculation. binding energy of K-pp increases, and its decay width becomes large. Non-rela. • SIDDHARTA constraint • KbarN potential is reconstructed in relativistic kinematics. Semi-rela. -ΓπYN [MeV]

  49. “Field picture or particle picture?” checked with Feshbach projection KbarN-πY coupled-channel energy-independent potential Feshbach projection KbarN single-channel energy-dependent potential B Γ/2 Full ccCSM 51 16 “Answer” ccCSM+Feshbach 49 17 • Field picture EKN = -BK 46 27 • Particle picture EKN = -BKN/2 45 29 • Λ*-fixed EKN = E(Λ*) Use Akaishi-Yamazaki potential

  50. Realization of self-consistency • Chiral SU(3) pot. (fπ=110 MeV, Martin) • Field picture • θ=22°, Dim=6400 Indicator of self-consistency Δ=|E(MB)Cal– E(MB)In| Indicator Δ [MeV] Self-consistent solution : BK-pp= 23.4 MeV, ΓπYN /2 = 9.5 MeV - Im E(MB)In [MeV] - Re E(MB)In [MeV]

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