Calculations of kaonic nuclei based on chiral meson-baryon coupled channel interaction models

Transcription

1 Calculations of kaonic nuclei based on chiral meson-baryon coupled channel interaction models J. Hrtánková, J. Mare² Nuclear Physics Institute, eº, Czech Republic 55th International Winter Meeting on Nuclear Physics Bormio, January 23-27, 217

2 Introduction 2 Introduction Interactions between K and nucleon(s) is topical but still not resolved problem Existence of the I = πσ resonance Λ(145) below K N threshold K N interaction attractive + strongly coupled to πσ channel strong K absorption Are there any (narrow) deeply bound K -nucler states? no decisive answer so far

3 Introduction Introduction Self-consistent calculations of K -nuclear quasi-bound states using the following chiral meson-baryon interaction models: Prague (P) (A. Cieply, J. Smejkal, Nucl. Phys. A 881 (212) 115) Kyoto-Munich (KM) (Y. Ikeda, T. Hyodo and W. Weise, Nucl. Phys. A 881 (212) 98) Murcia (M1 and M2) (Z. H. Guo and J. A. Oller, Phys. Rev. C 87 (213) 3522) Bonn (B2 and B4) (M. Mai and U.-G. Meiÿner, Nucl. Phys. A 9 (213) 51)

4 Model 4 Free-space K p amplitudes P KM M1 M2 B2 B Re F K - p (fm) 1.5 Im F K - p (fm) s 1/2 (MeV) s 1/2 (MeV) Fig.1: Energy dependence of real (left) and imaginary (right) parts of free-space K p amplitudes in considered models.

5 Model 5 Free-space K n amplitudes P KM M1 M2 B2 B Re F K - n (fm) 1.5 Im F K - n (fm) s 1/2 (MeV) s 1/2 (MeV) Fig.2:Energy dependence of real (left) and imaginary (right) parts of free-space K n amplitudes in considered models.

6 Model Model Klein-Gordon equation for K [ ω K 2 + ] 2 mk 2 Π K ( p K, ω K, ρ) φ K = complex energy ω K = m K B K iγ K /2 V C = ω K V C Self-energy operator Π K = 2Re(ω K )V (1) = 4π K s ( 1 F m N 2 ρ p + F 1 ( 1 2 ρ p + ρ n )), F and F 1 isospin and 1 scattering amplitudes from a chiral meson-baryon interaction model Nucleus described within an RMF model Static self-consistent calculations core polarization eect up to 5 MeV in K binding energies (D. Gazda, J. Mare², Nucl. Phys. A 881, 159 (212))

7 Model Model Free space amplitudes in-medium amplitudes - WRW method (T. Wass, M. Rho, W. Weise, Nucl. Phys. A 617 (1997) 449) F 1 = F K n( s) 1 + 1ξ s 4 k F m N K n( s)ρ, [2F K F = p( s) F K n( s)] 1 + 1ξ s 4 k [2F m N K p( s) F K n( s)]ρ where ξ k = 9π p 2 f 4 P + Pauli + SE model dt t exp(iqt)j 1 2 (t), q = ω 1 p K 2 m 2. f K (A. Cieply, J. Smejkal, Nucl. Phys. A 881 (212) 115) F ij (p, p ; s) = g i (p)g j (p ) 4πf i f j G i ( s; ρ) = 1 M i f 2 i s Ω i (ρ) M i M j [ (1 C( s) G( s) 1 ) C( s) ] s ij d 3 p g 2 i (p) (2π) 3 p 2 i p 2 Π i ( s, p; ρ) + i.,

8 Model 8 In-medium modied K N amplitudes P model Re f K - N (fm) free WRW Pauli Pauli + SE Im f K - N (fm) s 1/2 (MeV) s 1/2 (MeV) Fig.3: Energy dependence of reduced free-space (dotted line) f = 1 2 (f K p + f K n) amplitude compared with WRW modied amplitude (solid line), Pauli (dashed line), and Pauli + SE (dot-dashed line) modied amplitude for ρ =.17 fm 3 in the P model.

9 Model 9 Energy dependence K N amplitudes are a function of s (s = (E N +E K ) 2 ( p N + p K ) 2 ) K N cms frame K -nucleus frame p N + p K (A. Cieplý, E. Friedman, A. Gal, D. Gazda, J. Mare² PLB 72 (211) 42) Low-density limit δ s as ρ where δ s = s E th [ ( ) 2/3 ( ) ] 1/3 ρ s = Eth B N ρ ξ ρ ρ ρ N B K +23 +V C +ξ ρ max ρ ρ K ReV K (r), max where B N = 8.5 MeV and ξ N(K ) = m N(K )/(m N + m K ).

10 Results 1 Energies probed in the calculations -2 P KM M1 M2 E - E th (MeV) ρ/ρ Fig.4: Subthreshold energies probed in the 16 O+K nucleus as a function of relative density ρ/ρ, calculated self-consistently using K N amplitudes in the P, KM, M1, and M2 models.

11 Results K 1s binding energies and widths 1 1s Ca Zr Pb 1 P KM M1 M2 B K C O Γ K C O Ca Pb Zr A 2/ A 2/3 Fig.5: 1s K binding energies (left) and corresponding widths (right) in various nuclei calculated self-consistently in the P, KM, M1, and M2 models.

12 Results Multinucleon processes K interactions with two and more nucleons - recent analysis of kaonic atom data including branching ratios of K absorption by Friedman and Gal (NPA 959 (217) 66) 2Re(ω K )V (2) K = 4πB( ρ ρ ) α ρ, (1) where B is a complex amplitude and α is positive Only P and KM models preferred by the analysis (P1, KM1 for α = 1 and P2, KM2 for α = 2) K NN ΣN dominant K absorption mode in the nuclear interior ImB multiplied by a kinematical suppression factor f ΣN

13 Results 13 Multinucleon processes Total K optical potential V K = V (1) + K V (2) K Experiments with kaonic atoms probe the K optical potential (mainly its imaginary part) up to 5% of ρ We consider two limiting cases for V (2) K in our calculations form (1) in the entire nucleus full density option (FD) x V (2) at constant value K V (2) (.5ρ K ) for ρ(r).5ρ half density limit (HD)

14 Results 14 Total K optical potential ReV K KM1 FD HD ImV K HD FD KM r (fm) r (fm) Fig.6: The real and imaginary parts of the K optical potential in the 28 Pb+K nucleus, calculated self-consistently in the KM1 model for two versions of the K multinucleon potential. The single-nucleon K potential (green solid line) calculated in the corresponding model is shown for comparison.

15 Results 15 Contributions to the total K optical potential ReV K N total KM1 +N ImV K N KM1 + N total r (fm) r (fm) Fig.7: The respective contributions from K N and K NN potentials to the total real and imaginary K optical potential in the 28 Pb+K nucleus, calculated self-consistently in the KM1 model and FD variant. The single-nucleon K potential (green solid line) calculated in the corresponding model is shown for comparison.

16 Results K 1s binding energies and widths B K s C O Ca Zr HD FD Pb Γ K KM1 FD C O Ca Zr Pb HD A 2/ A 2/3 Fig.8: 1s K binding energies (left) and corresponding widths (right) in various nuclei calculated self-consistently for two options of V (2) in the KM1 model. K

17 Results 17 Ratios of K absorption in medium ImV K - (ImV K -) / ImV K - (1) (2) ρ/ρ HD FD KM1 N ImV K - (ImV K -) / ImV K - (1) (2) KM1 P1 KM2 P r (fm) r (fm) Fig.9: Ratios of ImV (1) and K ImV (2) potentials to the total K K imaginary potential ImV K as a function of radius, calculated self-consistently for 28 Pb+K system in the KM1 model and dierent option for the K multinucleon potential (left) and the comparison of these ratios calculated in dierent meson-baryon interaction models for FD option (right). The vertical lines denoting 15% of ρ are shown for comparison.

18 Conclusions Conclusions Calculations of K -nuclear quasi-bound states in various nuclei K single-nucleon potentials based on chiral meson-baryon interaction models yield: large model dependence of K binding energies small K widths Sizeable contribution from K multinucleon interactions inside the nucleus: K widths signicantly larger than binding energies, if ever bound

19 Conclusions Thank you for your attention! 19

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21 backup 21 K optical potential -2 Im Im -2 V K P - WRW Im Re Re Re Im Re P - Pauli + SE E th s 1/ V K r (fm) r (fm) Fig.1: K optical potential in 4 Ca calculated self-consistently for in-medium s and at threshold E th using WRW modied amplitudes in the P NLO model compared with P NLO + Pauli + SE model.

22 backup 22 K optical potential ReV K KM1 FD TR HD ImV K HD TR FD KM r (fm) r (fm) Fig.11: 1s K binding energies (left) and corresponding widths (right) in various nuclei calculated self-consistently for dierent options of V (2) in the KM1 model. K

23 backup K 1s binding energies and widths B K s C O Ca Zr HD FD Pb Γ K KM2 FD C Ca Zr Pb O HD A 2/ A 2/3 Fig.12: 1s K binding energies (left) and corresponding widths (right) in various nuclei calculated self-consistently for two options of V (2) in the KM2 model. K

24 backup 24 Multinucleon processes Table : Values of the complex amplitude B and exponent α used to evaluate V (2) K for all chiral meson-baryon interaction models considered in this work. B2 B4 M1 M2 P1 KM1 P2 KM2 α ReB (fm) ImB (fm)