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K K N bar N bar Nara Women's University Contents Contents Properties of (1405) state Meson exchange K bar N potential

  1. 中間子交換模型によるK K N相互作用 相互作用 bar N 中間子交換模型による bar 佐々木 健志 Nara Women's University

  2. Contents Contents ✗ Properties of Λ (1405) state ✗ Meson exchange K bar N potential ✗ Comparison with the chiral amplitudes ✗ Energy dependence of the potential ✗ Conclusion

  3. Λ (1405)? What is the Λ (1405)? What is the According to the PDG Most-established resonance with four-stars in PDG The Λ (1405) can be observed directly only as a resonance bump in the ( Σπ ) 0 subsystem in final states of production experiments. Theoretical interpretation ??? Quark model fails to reproduce splitting between Λ (1405) and Λ (1520) 3q state, meson-baryon system, two pole ? S.Capstick '89

  4. Λ (1405)? What is the Λ (1405)? What is the According to the PDG Most-established resonance with four-stars rating by PDG The Λ (1405) can be observed directly only as a resonance bump in the ( Σπ ) 0 subsystem in final states of production experiments.

  5. Experimental view of Λ (1405) M.H.Alston '61 R.J.Hemingway '85 D.W.Thomas '73 Asymmetric shape of the resonance bump not well fitted by a Breit-Wigner resonance function Direct evidence for J p =1/2 -

  6. Theoretical interpretation of Λ (1405) Three quark state Meson + Baryon E.A.Veit '85 S.Capstick '89 N.Isgur '78 N.Kaiser '95 Λ (1405) is dominated by the meson-baryon terms in the wavefunctions.

  7. bar N interaction bar Juelich K N interaction Juelich K

  8. bar N interaction bar The Juelich K N interaction The Juelich K A.Muller-Groeling-NPA513(1990)557 (R.Buttgen-NPA506(1990)586) ➔ Meson (hadron) exchange model ➔ K bar N, πΣ, πΛ channels are considered (Coupled channel approach) Diagrams Potential is constructed by small number of vertices Main contribution comes from the vector meson exchange

  9. bar N interaction bar The Juelich K N interaction The Juelich K Flavor SU(3) symmetry is assumed Hamiltonians for meson-baryon couplings Determined by baryon-baryon scattering Hamiltonians for meson-meson couplings Parameters are determined by KN scattering

  10. bar N interaction bar The Juelich K N interaction The Juelich K Cross sections Invariant mass distribution Peak around 1400MeV Λ (1405) state can be seen at proper position without the pole graph in V. It is predicted as the quasi-bound state of K bar N. Consistent with experimental data

  11. bar N potential bar Phenomenological K N potential Phenomenological K

  12. Y. Akaishi and T. Yamazaki, PRC52(2002)044005 Phenomenological AY potential Phenomenological AY potential Ansatz The Λ (1405) resonance state is the I= 0 1 s bound state of K bar N Regarding 1. 1 s level shift of kaonic hydrogen atom 2. Martin's K bar N scattering lengths 3. Binding energy and width of Λ(1405) K bar N- πΣ coupled channel with I=0 K bar N πΣ K bar N πΣ Equivalent single channel potential Various kaonic nuclear states with large binding energy and high density

  13. Y. Akaishi and T. Yamazaki, PRC52(2002)044005 Phenomenological AY potential Phenomenological AY potential Points ✗ Λ (1405) ansatz (B.E = 27MeV) ✗ Energy independent potential ✗ Omission of the diagonal πΣ -channel interaction ✗ Compact object 1.36fm between K bar and N (rms distance)

  14. Chiral effective theory Chiral effective theory

  15. D. Jido et al NPA725(2003)181 Chiral effective theory Chiral effective theory T. Hyodo-PRC77(2008)035204 Seagull (Tomozawa-Weinberg) term from chiral effective lagrangian T-matrix is solved algebraically(on-shell treatment) Choice of decay constant f and regularization mass in the loop function G Evidence of meson-baryon state with natural subtraction constant Two poles near Λ (1405)

  16. Chiral effective theory Chiral effective theory Cross sections Invariant mass distribution Peak around 1400MeV The resonance shape is generated as an interference of two poles Consistent with experimental data

  17. Roles of vector meson exchange potential Roles of vector meson exchange potential

  18. Vector meson exchange potentials Vector meson exchange potentials K bar N to K bar N PPV couplings N K N K Coupling constants of PPV vertex L ppv = g Tr [ V  [ P , ∂  P ] ] ρ ω N K N Empirical V −> PP decay width and SU(3) K K bar N to πΛ K bar N to πΣ Λ π BBV couplings Σ π Coupling constants of BBV vertex Κ ∗ Κ ∗   f L BBV =  q  ] V  B B [ g  2M  N K N K πΣ to πΛ πΣ to πΣ Vector coupling Tensor coupling Λ π Σ π The f/g are taken from the Bonn potential and SU(3). ρ ρ The strength of g is determined by following way. Σ Σ π π

  19. Vector meson exchange potentials Vector meson exchange potentials K bar N to K bar N PPV couplings N K N K Coupling constants of PPV vertex L ppv = g Tr [ V  [ P , ∂  P ] ] ρ ω N K N Empirical ρ−>ππ decay width and SU(3) K K bar N to πΛ K bar N to πΣ Λ π BBV couplings Σ π Coupling constants of BBV vertex Κ ∗ Κ ∗   f L BBV =  q  ] V  B B [ g  2M  N K N K πΣ to πΛ πΣ to πΣ Vector coupling Tensor coupling Λ π Σ π The f and g are taken from the Bonn potential ρ ρ Σ Σ π π

  20. Comparison with the TW term Comparison with the TW term Vector meson exchange Tomozawa-Weinberg term Β P Β P Vector dominance ansatz q → 0 V B P B P 2  ∗ p f  p i     ∗  g  g    f  q    q    ∗ p f  p i  2M   q q → 0 m They would be the same contribution at q=0 limit In the q=0 limit E B ≃ M B ,  s = M B  m P , t = 0 Vector meson exchange Tomozawa-Weinberg term V th = g 1 g 2 V th = C f 2 m 2

  21. K.Kawarabayashi-PRL16(1966)255 Threshold behaviors Riazuddin-PRev147(1966)1071 Vector meson T-W ratio KN to KN I=0 -0.839 -0.750 1.119 I=1 -0.270 -0.250 1.081 KN to πΣ I=0 0.264 0.306 0.862 I=1 0.213 0.250 0.852 KN to πΛ I=1 0.261 0.306 0.851 πΣ to πΣ I=0 -1.153 -1.000 1.153 I=1 -0.569 -0.500 1.138 Deviation from SU(3) value of K * →K π decay constant 2 = 2 f 2 g V 2 KSRF relation:m V Effect of form-factor ? 2  2 = exp  2 2 − m 2 2  2 =  − q 2 = F  q F  q F  q 2  q 2  2  q 2   2 − 0.78 2 F   0 = 1.5 = 0.73 1.5 2

  22. Vector meson exchange potentials Vector meson exchange potentials K bar N to K bar N PPV couplings N K N K Cutoff parameters Λ NS : P bar P V NS coupling vertices ρ ω Λ S : K bar π V S coupling vertex N K N K K bar N to πΛ K bar N to πΣ BBV couplings Λ π Σ π Cutoff parameters Κ ∗ Κ ∗ Λ NS : BB V NS coupling vertices Λ S : NY V S coupling vertices N K N K πΣ to πΛ πΣ to πΣ Monopole or Gaussian form factors are employed Λ π Σ π 2  2 = exp  2 2  − q 2 = F  q F  q ρ 2  q 2 ρ   Σ Σ π π

  23. Results of the vector meson exchange Results of the vector meson exchange Scattering cross sections compared with chiral unitary calculations These results are obtained by changing the cutoffs for each vertex The vector meson plays a crucial role in the K bar N system

  24. bar N interaction bar Comparison with the Julich K N interaction Comparison with the Julich K Cross sections Σ (1385) cotribution

  25. Comparison with the chiral effective theory Comparison with the chiral effective theory Cross sections

  26. Results of the vector meson exchange Results of the vector meson exchange Scattering amplitudes Amplitude(I=0 KN) Amplitude(I=0 pS) Amplitude(I=1 KN) 0.8 4 1.5 3 0.6 1 2 1 0.5 0.4 0 0 0.2 -1 -0.5 -2 0 1320 1360 1400 1440 1320 1360 1400 1440 1320 1360 1400 1440 ✗ The KSRF corrected coupling constants are used in calculation ✗ Cutoff parameters are : Λ NS =1.5GeV, Λ S =2.2GeV This model is similar to the chiral unitary model

  27. Results of the vector meson exchange Results of the vector meson exchange Scattering amplitudes ✗ The KSRF corrected coupling constants are used in calculation ✗ Cutoff parameters are Λ NS =1.5GeV and Λ S =2.2GeV. This model is similar to the chiral unitary model Scattering lengths are reproduced fairly well

  28. Invariant mass plot Invariant mass plot πΣ scattering amplitude 4000 No peak is seen in amplitude by meson exchange model 3000 1 2000 0.8 0.6 1000 0.4 0 1.4 0.2 Effect of πΣ amplitude 0 1300 1350 1400 1450 1500 Pole in T-matrix z = 1388 − 96 i MeV T. Hyodo-PRC77(2008)035204 Energy dependent potential creates the πΣ resonance pole? D.Jido[ 特定領域研究会 2006]

  29. bar N potential bar Energy dependence of K N potential Energy dependence of K

  30. General form of vector meson exchange potential General form of vector meson exchange potential The t-matrix for the meson-baryon scattering Central potential (spin independent) L-S potential (spin dependent) The functions, A and B, for the vector meson exchange are

  31. Form of vector meson exchange potential Form of vector meson exchange potential Options of momentum configurations Nonlocal (and energy dependent potential) Local potential Same potential at the case of k=0

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