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Application of coupled-channel Complex Scaling Method to the K bar N -πY system

Application of coupled-channel Complex Scaling Method to the K bar N -πY system. A. Doté ( KEK Theory center / IPNS/ J-PARC branch) T. Inoue (Nihon univ .) T. Myo (Osaka Inst. Tech. univ .). Introduction c oupled-channel Complex Scaling Method For resonance states

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Application of coupled-channel Complex Scaling Method to the K bar N -πY system

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  1. Application of coupled-channel Complex Scaling Method to the KbarN-πY system A. Doté(KEK Theory center / IPNS/ J-PARC branch) T. Inoue (Nihon univ.) T. Myo (Osaka Inst. Tech. univ.) • Introduction • coupled-channel Complex Scaling Method • For resonance states • For scattering problem • Set up for the calculation of KbarN-πY system • “KSW-type potential” • Kinematics / Non-rela. Approximation • Results • I=0 channel: Scattering amplitude / Resonance pole • I=1 channel: Scattering amplitude • Summary and future plans A. Dote, T. Inoue, T. Myo, Nucl. Phys. A912, 66-101 (2013) International workshop “Inelastic Reactions in Light Nuclei”, 09.Oct.’13 @ IIAS, The Hebrew univ., Jerusalem, Israel

  2. 1. Introduction

  3. Kaonic nuclei = Nuclear system with Anti-kaon“Kbar” Attractive KbarN interaction! Excited hyperon Λ(1405) = a quasi-bound state of K- and proton s ubar • Difficult to explain by naive 3-quark model • Consistent with “repulsive nature” indicated by KbarN scattering length and 1s level shift of kaonichydron atom u s KbarN threshold = 1435MeV d u d u T.Hyodoand D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)

  4. Kaonic nuclei = Nuclear system with Anti-kaon“Kbar” Attractive KbarN interaction! Excited hyperon Λ(1405) = a quasi-bound state of K- and proton • Difficult to explain by naive 3-quark model • Consistent with “repulsive nature” indicated by KbarN scattering length and 1s level shift of kaonichydron atom Anti-kaon can be deeply bound and/or form dense state? • Antisymmetrized Molecular Dynamics with a phenomenological KbarN potential • Anti-kaon bound in light nuclei with ~ 100MeV binding energy • Dense matter (average density = 2～4 ρ0) • Relativistic Mean Field applied to medium to heavy nuclei with anti-kaons • AMD studyA. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313. • RMF studyD. Gazda, E. Friedman, A. Gal, J. Mares, PRC 76 (2007) 055204; PRC 77 (2008) 045206;A. Cieply, E. Friedman, A. Gal, D. Gazda, J. Mares, PRC 84 (2011) 045206.T. Muto, T. Maruyama, T. Tatsumi, PRC 79 (2009) 035207.

  5. Kaonic nuclei = Nuclear system with Anti-kaon“Kbar” Attractive KbarN interaction! Excited hyperon Λ(1405) = a quasi-bound state of K- and proton • Difficult to explain by naive 3-quark model • Consistent with “repulsive nature” indicated by KbarN scattering length and 1s level shift of kaonichydron atom Anti-kaon can be deeply bound and/or form dense state? • Antisymmetrized Molecular Dynamics with a phenomenological KbarN potential • Anti-kaon bound in light nuclei with ~ 100MeV binding energy • Dense matter (average density = 2～4 ρ0) Prototype of kaonic nuclei = “K-pp” • Relativistic Mean Field applied to medium to heavy nuclei with anti-kaons A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009) • Variational calculation with Gaussian base • Av18 NN potential • Chiral SU(3)-based KbarN potential B. E. ~ 20MeV, Γ = 40 ~70MeV pp distance ~ 2.1 fm

  6. Theoretical studies of K-pp “K-pp”= Prototype of Kbarnuclei (KbarNN, Jp=1/2-, T=1/2) • Doté, Hyodo, Weise PRC79, 014003(2009) • Variational with a chiral SU(3)-based KbarN potential • Akaishi, Yamazaki PRC76, 045201(2007) • ATMS with a phenomenological KbarN potential • Ikeda, Sato PRC76, 035203(2007) • Faddeev with a chiral SU(3)-derived KbarN potential • Shevchenko, Gal, Mares PRC76, 044004(2007) • Faddeev with a phenomenological KbarN potential • Barnea, Gal, LivertsPLB712, 132(2012) • Hyperspherical harmonics with a chiral SU(3)-based KbarNpotential • Wycech, Green PRC79, 014001(2009) • Variational with a phenomenological KbarN potential (with p-wave) • Arai, Yasui, Oka/Uchino, Hyodo, OkaPTP119, 103(2008) • Λ* nuclei model /PTPS 186, 240(2010) • Nishikawa, Kondo PRC77, 055202(2008) • Skyrme model All studies predict that K-pp can be bound!

  7. Theoretical studies of K-pp “K-pp”= Prototype of Kbarnuclei (KbarNN, Jp=1/2-, T=1/2) • Doté, Hyodo, WeisePRC79, 014003(2009) • Variational with a chiral SU(3)-based KbarN potential • Akaishi, YamazakiPRC76, 045201(2007) • ATMS with a phenomenological KbarN potential • Ikeda, SatoPRC76, 035203(2007) • Faddeev with a chiral SU(3)-derived KbarN potential • Shevchenko, Gal, MaresPRC76, 044004(2007) • Faddeev with a phenomenological KbarN potential • Barnea, Gal, LivertsPLB712, 132(2012) • Hyperspherical harmonics with a chiral SU(3)-based KbarNpotential • Wycech, Green PRC79, 014001(2009) • Variational with a phenomenological KbarN potential (with p-wave) • Arai, Yasui, Oka/Uchino, Hyodo, OkaPTP119, 103(2008) • Λ* nuclei model /PTPS 186, 240(2010) • Nishikawa, Kondo PRC77, 055202(2008) • Skyrme model All studies predict that K-pp can be bound!

  8. Typical results of theoretical studies of K-pp Width (KbarNN→πYN) [MeV] Barnea, Gal, Liverts [5] (HH, Chiral SU(3)) Doté, Hyodo, Weise [1] (Variational, Chiral SU(3)) Akaishi, Yamazaki [2] (Variational, Phenomenological) Shevchenko, Gal, Mares [3] (Faddeev, Phenomenological) - B.E. [MeV] Ikeda, Sato [4] (Faddeev, Chiral SU(3)) -100 MeV Exp. : DISTO [7] if K-pp bound state. Exp. : FINUDA [6] if K-pp bound state. Using S-wave KbarN potential constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc. [1] PRC79, 014003 (2009) [5] PLB94, 712 (2012) [2] PRC76, 045201 (2007) [3] PRC76, 044004 (2007) [4] PRC76, 035203 (2007) [6] PRL94, 212303 (2005) [7] PRL104, 132502 (2010)

  9. Typical results of theoretical studies of K-pp Width (KbarNN→πYN) [MeV] Kbar+N+N threshold Barnea, Gal, Liverts [5] (HH, Chiral SU(3)) Doté, Hyodo, Weise [1] (Variational, Chiral SU(3)) Akaishi, Yamazaki [2] (Variational, Phenomenological) Shevchenko, Gal, Mares [3] (Faddeev, Phenomenological) - B.E. [MeV] Ikeda, Sato [4] (Faddeev, Chiral SU(3)) π+Σ+N threshold Exp. : DISTO [7] if K-pp bound state. Exp. : FINUDA [6] if K-pp bound state. From theoretical viewpoint, K-pp exists between Kbar-N-N and π-Σ-N thresholds! Using S-wave KbarN potential constrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc. [1] PRC79, 014003 (2009) [5] PLB94, 712 (2012) [2] PRC76, 045201 (2007) [3] PRC76, 044004 (2007) [4] PRC76, 035203 (2007) [6] PRL94, 212303 (2005) [7] PRL104, 132502 (2010)

  10. Key points to study kaonic nuclei • Coupling of KbarN and πY • Bound below KbarN threshold, but a resonant state above πY threshold. • Their nature? Kbar + N + N π + Σ + N “coupled-channel Complex Scaling Method” 1. Consider a coupled-channel problem “Kbar N N” 2. Treat resonant states adequately. Similar to the bound-state calculation 3. Obtain the wave function to help the analysis of the state. 4. Confirmed that CSM works well on many-body systems. KbarN (-πY) interaction?? “Chiral SU(3)-based potential” … Anti-kaon = Nambu-Goldstone boson

  11. 2. Complex Scaling Method Λ(1405) = a building block of kaonic nuclei K- Proton • For Resonance states • For Scattering problem

  12. Λ(1405) with c.c. Complex Scaling Method Kbar + N 1435 Λ(1405) π + Σ 1332 [MeV] Kbar (Jπ=0-, T=1/2) π (Jπ=0-, T=1) L=0 L=0 N (Jπ=1/2+, T=1/2) Σ (Jπ=1/2+, T=1) KbarN-πΣ coupled system with s-wave and isospin-0 state

  13. Complex Scaling Method for Resonance Complex rotation of coordinate (Complex scaling) By Complex scaling, … • Resonant state Divergent κ, γ: real, >0 Damping under some condition

  14. Complex Scaling Method for Resonance Complex rotation of coordinate (Complex scaling) By Complex scaling, • Resonance wave function: divergent function ⇒ damping function • Resonance energy (pole position) doesn’t change. DiagonalizeHθ with Gaussian base, • ABC theorem • “The energy of bound and resonant states is independent of scaling angle θ.” • † J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269. • E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280 Boundary condition is the same as that for a bound state. we can obtain resonant states, in the same way as bound states!

  15. Complex Scaling Method for Resonance Test calculation with a phenomenological potential Akaishi-Yamazaki potential† • Local Gaussian form • Energy independent πΣ KbarN [MeV] KbarN πΣ q=30 deg. B. E. (KbarN) = 28.2 MeV Γ = 40.0 MeV pScontinuum KbarN continuum … Nominal position of Λ(1405) †Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)

  16. 2. Complex Scaling Method Calculation of KbarN scattering amplitude • For Resonance states • For Scattering problem

  17. Calc. of scattering amplitude with CSM With help of the CSM, all problems for bound, resonant and scattering states can be treated with Gaussian base!

  18. Calc. of scattering amplitude with CSM A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007) 1. Separate incoming wave • Unknown • Non square-integrable 2. Complex scaling r → reiθ square-integrable for 0 < θ < π Expanding with square-integrable basis function (ex: Gaussian basis) Cauchy theorem 3. Calculate scattering amplitude with help of Cauchy theorem Scattered wave function along reiθ is known! r eiθ By Cauchy theorem, we can obtain as Born term is OK! r Scattered part is unknown. 0

  19. Calc. of scattering amplitude with CSM A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007) Scattering problem can be solved as bound-state problem by matrix calculation! Equation to be solved: can be expanded with Gaussian basis square-integrable for 0 < θ < π Linear equation to be solved with matrix calculation!

  20. 3. Set up for the calculation of KbarN-πYsystem • “KSW-type potential” … Chiral SU(3)-based • Kinematics • Non-rela. approximation of KSW-type potential

  21. “KSW-type potential” … chiral-SU(3) based Effective Chiral Lagrangian Pseudopotential • Delta-function type (→ Yukawa / separable type) • Up to order q2 N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995) KbarN πΣ “KSW-type potential” • Local Gaussian form in r-space • Weinberg-Tomozawa term • Energy dependence → Easy to handle in many-body calculation with Gaussian base a(I)ij: range parameter [fm] →　Relative strength between channels determined by the SU(3) algebra ←ChiralSU(3) theory Constrained by KbarN scattering length (Martin’s value) aKN(I=0) = -1.70+i0.67fm, aKN(I=1)= 0.37+i0.60fm A.D.Martin, NPB179, 33(1979)

  22. Kinematics 1. Non-relativistic Reduced mass 2. Semi-relativistic Reduced energy

  23. Non-rela. approximation of KSW potential - Original • Normalized Gaussian • Range parameter • for coupling potential → (NRv1)Non-rela. approx. version 1 Comparison of the flux factor for differential cross section between non-rela. and rela. → (NRv2)Non-rela. approx. version 2 @ non-rela. limit (small p2)

  24. Kinematics Potential Semi-rela. - Original (NRv1)Non-rela. approx. ver. 1 Non-rela. (NRv2)Non-rela. approx. ver. 2

  25. 4. Result Using Chiral SU(3) potential “KSW-type” … r-space, Gaussian form, Energy-dependent • I=0 channel • I=0 KbarN scattering lengthRange parameters of KSW-type potential • Scattering amplitude • Resonance pole … wave function and size

  26. I=0 KbarN scattering length Range parameters of KSW-type potential fπ=110 MeV dKN,KN dπΣ,πΣ Re aI=0KN ImaI=0KN Range parameters: • Found sets of range parameters (dKN,KN, dπΣ,πΣ) to reproduce the Martin’s value. • Two sets found in semi-rela. case. (SR-A, SR-B) Pion decay constant “fπ” as a parameter fπ = 90 ~ 120 MeV

  27. Scattering amplitude … Non-rela. / KSW-type NRv1 fπ=110 MeV πΣ KbarN πΣ KbarN Im Re KbarN → KbarN πΣ→ πΣ Resonancestructure at 1413 MeV Resonancestructure at 1405 MeV Resonance structure appears in KbarN and πΣchannels.

  28. Scattering amplitude … Non-rela. / KSW-type NRv2 fπ=110 MeV πΣ KbarN πΣ KbarN Im Re KbarN → KbarN πΣ→ πΣ Resonancestructure at 1416 MeV Resonancestructure at 1408 MeV Essentially same as NRv1 case.

  29. Scattering amplitude … Semi-rela. / KSW-type SR-A fπ=110 MeV πΣ KbarN πΣ KbarN Im Re KbarN → KbarN πΣ→ πΣ Resonancestructure at 1410 MeV Resonancestructure at 1398 MeV Qualitatively similar to non-rela. amplitudes.

  30. Scattering amplitude … Semi-rela. / KSW-type SR-B fπ=110 MeV (Another set of SR) πΣ KbarN πΣ KbarN ??? Im Re KbarN → KbarN πΣ→ πΣ Resonancestructure at 1419 MeV Resonancestructure at 1421 MeV Very different behavior of πΣ amplitude from other cases.

  31. Pole position of the resonance fπ=90 - 120MeV M [MeV] 120 120 -Γ / 2 [MeV] -Γ / 2 [MeV] 120 90 90 120 90 90 Complex energy plane

  32. Pole position of the resonance fπ=90 - 120MeV M [MeV] 120 Non-rela. 120 -Γ / 2 [MeV] -Γ / 2 [MeV] 120 90 90 120 (M, Γ/2) = (1418.2 ± 1.6, 21.4 ± 4.7) … NRv1 (1418.9 ± 1.1, 18.6 ± 4.6) … NRv2 90 90

  33. Pole position of the resonance fπ=90 - 120MeV M [MeV] 120 120 -Γ / 2 [MeV] -Γ / 2 [MeV] 120 90 (M, Γ/2) = (1420.5 ± 3, 24.5 ± 2) … SR-A 90 120 Semi-rela. 90 90

  34. Pole position of the resonance fπ=90 - 120MeV M [MeV] Different behavior from other cases 120 120 -Γ / 2 [MeV] -Γ / 2 [MeV] 120 90 Semi-rela. (Another set) 90 120 90 90 (M, Γ/2) = (1419.5 ± 0.5, 13.0 ± 1.4) … SR-B

  35. “Wave function” of the resonance pole fπ=110 MeV θ=30° Semi rela. (SR-A) Non-rela. (NRv2) πΣcomponent also localized due to Complex scaling. ? Somehow small. cf) 1.9 fm @ M = 1423 MeV† (B = 12 MeV) - Difference of binding energy? - Different definition of pole? Gamow state or bound state †A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009) Mean distance between meson and baryon m NR: ～ 1.3 - 0.3 i [fm] SR: ～ 1.2 - 0.5 i [fm] B

  36. 4. Result Using Chiral SU(3) potential (KSW-type) … r-space, Gaussian form, Energy-dependent • I=1 channel • Range parameters of the KSW-type potential • Scattering amplitude

  37. I=1 channel … KbarN - πΣ - πΛ Potential … Weinberg-Tomozawa term only Data : aI=1KbarN = 0.37 + i 0.60 fm (A. D. Martin)† Range parameters: † A. D. Martin, Nucl. Phys. B 179, 33 (1981)

  38. I=1 channel … KbarN - πΣ - πΛ Potential … Weinberg-Tomozawa term only Data : aI=1KbarN = 0.37 + i 0.60 fm (A. D. Martin)† Range parameters: × SU(3) C.G. coefficients for I=1 channel ×

  39. I=1 channel … KbarN - πΣ - πΛ Potential … Weinberg-Tomozawa term only Data : aI=1KbarN = 0.37 + i 0.60 fm (A. D. Martin)† Range parameters: × SU(3) C.G. coefficients for I=1 channel ×

  40. I=1 channel … KbarN - πΣ - πΛ Potential … Weinberg-Tomozawa term only Data : aI=1KbarN = 0.37 + i 0.60 fm (A. D. Martin)† “Isospin symmetric choice” Ref.) chiral unitary model Range parameters: × SU(3) C.G. coefficients for I=1 channel ×

  41. I=1 channel … KbarN - πΣ - πΛ • “Iso-symmetric choice” : {dKN,KN, dπΣ,πΣ} fixed to I=0 channel ones • Search dKN,πΣ to reproduce Re or ImaI=1KbarN of Matin’s value. A. D. Martin : aI=1KbarN = 0.37 + i 0.60 fm (I=0 one) (Search) (fit) (fit) (fit) (fit) When Re aI=1KbarN is reproduced, ImaI=1KbarN deviates largely from Martin’s value. Difficult to reproduce Re aI=1KbarN Martin’s value within our model.

  42. I=1 channel … KbarN - πΣ - πΛ fπ=110 MeV • Search dKN,πΣ to reproduce ImaI=1KbarN of Matin’s value. NRv2 (c): aI=1KbarN = 0.657 + i 0.599 fm KbarN πΣ πΛ SR-A (c): aI=1KbarN = 0.659 + i 0.600 fm KbarN πΣ πΛ

  43. I=1 channel … KbarN - πΣ - πΛ • Only dKN,KN fixed to I=0 channel onesIn a study with separable potential, the cutoff parameter for KbarN is not so different between I=0 and 1 channels.† • Search {dπΣ,πΣ, dKN,πΣ} to reproduce Re and ImaI=1KbarN of Matin’s value. † Y. Ikeda and T. Sato, PRC 76, 035203 (2007) (I=0 one) (Search) (fit) (fit) (fit) (fit) When the only dKN,KNis fixed iso-symmetrically, we can find a set of {dπΣ,πΣ, dKN,πΣ} to reproduce simultaneously Re and Im of Martin’s value

  44. I=1 channel … KbarN - πΣ - πΛ fπ=110 MeV • Search {dπΣ,πΣ, dKN,πΣ} to reproduce Re and ImaI=1KbarN of Matin’s value. NRv2 (a): aI=1KbarN = 0.376 + i 0.606 fm KbarN πΣ πΛ A narrow resonance exists at a few MeV below πΣ threshold SR-A (a): aI=1KbarN = 0.375 + i 0.605 fm KbarN πΣ πΛ πΣ repulsive ???

  45. 5. Summary and Future plan

  46. 5. Summary KbarN-πY system is essential for the study of Kbar nuclear system which is expected to be an exotic nuclear system with strangeness. Scattering and resonant states of KbarN-πY system is studied with a coupled-channel Complex Scaling Method using a chiral SU(3) potential • A Chiral SU(3) potential “KSW-type” … r-space, Gaussian form, energy dependence • Calculated scattering amplitude with help of CSMScattering states as well as resonant and bound states are treated with Gaussian base. Non-rela. / Semi-rela. kinematics and two types of non-rela. approximation of KSW-type potential are tried. • Determined by theKbarN scattering length obtained by Martin’s analysis of old data. • fπ dependence (fπ = 90 ～ 120MeV) • Found two sets of range parameters in SR case.

  47. 5. Summary and future plans • I=0 channel (KbarN-πΣ) • Pole position • “Size” of the I=0 pole state (M, Γ/2) NRv1 : (1418.2 ± 1.6, 21.4 ± 4.7) NRv2 : (1418.9 ± 1.1, 18.6 ± 4.6) SR-A : (1420.5 ± 0.5, 24.5 ± 2) SR-B : (1419 ± 1, 13.0 ± 2) NR: ～ 1.3 – 0.3i fm SR: ～ 1.2 – 0.5i fm Another self-consistent solution NR: (~1360, 40~90), SR: (1350~1390, 30~100) … Lower pole of double pole? • I=1 channel (KbarN-πΣ-πΛ) • Difficult to reproduce Re aI=1KbarN of Martin’s value within our model,in case of “iso-symmetric choice” of range parameter. Future plans Three-body system (KbarNN-πYN); Updated data of K-p scattering length by SHIDDARTA

  48. Thank you very much! A. D. is thankful to Prof. Katō for his advice on the scattering-amplitude calculation in CSM, and to Dr. Hyodo for useful discussion.

  49. Backup slides

  50. Rough estimation of charge radius CM K- Distance p † Calculated from electric form factor with chiral unitary model T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)

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