Recent theoretical studies Akinobu Dot (KEK Theory Center, IPNS / - PowerPoint PPT Presentation

The simplest kaonic nucleus “K - pp ” － Recent theoretical studies － Akinobu Doté (KEK Theory Center, IPNS / J-PARC branch) Takashi Inoue (Nihon university) Takayuki Myo (Osaka Institute of Technology) 1. Introduction “K - pp ” investigated with 2. Fully coupled-channel Complex Scaling Method • Formalism • Chiral SU(3)-based potential • Self-consistency for energy-dependent potential in coupled-channel case 3. Result 4. Discussion 5. Summary and future prospects 『 2017 年度 KEK 理論センター J-PARC 分室活動 総括研究会』 , 02. Feb. ’18 @ IQBRC, Tokai, Ibaraki, Japan

1. Introduction  Kaonic nuclei = Nuclear system with K bar mesons  Strongly attractive K bar N potential Excited hyperon Λ(1405) = K bar N quasi-bound state T. Hyodo, D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012) Kaonic nuclei = Exotic system!? Y. Akaishi, T. Yamazaki, PRC65, 044005 (2002) Doorway to dense matter? A. Dote, H. Horiuchi, Y. Akaishi, T. Yamazaki, PRC70, 044313 (2004)

We are investigating ... A prototype of kaonic nuclei ... a bridge from Λ(1405) to general kaonic nuclei

Experimental search for K - pp • Deeply bound region (near πΣN threshold = 103 MeV below K bar NN threshold) FINUDA (2005), DISTO (2010), J-PARC E27 (2013) • Shallowly bound region (near K bar NN threshold) J-PARC E15 1 st run (2013) • No signal in bound region and more ... LEPS/SPring8 (2015) Clear evidence of K - pp bound state J-PARC E15 (2 nd run): Exclusive exp. 3 He(K - , Λp )n missing

Theoretical studies of “K - pp” • K - pp should be bound. B K-pp < 100 MeV J-PARC E27 Resonance between πΣN and K bar NN thresholds. DISTO Faddeev-AGS J-PARC E15 Pheno. pot. (E-indep.) FINUDA Variational (Gauss) Pheno. pot. (E-indep.) Variational (Gauss) • Binding energy depends Chial. pot. (E-dep.) Faddeev-AGS on potential type. Chiral pot. (E-dep.) K - pp binding energy [MeV] πΣN threshold K bar NN threshold

2. “K - pp” investigated with Fully coupled-channel Complex Scaling Method • Formalism • Chiral SU(3)-based potential • Self-consistency for energy-dependent potential in coupled-channel case Discussed with Harada-san, Shinmura-san, and Akaishi-san in J-PARC branch activities

According to early studies ... Prototype system of kaonic nuclei “K - pp” • Akaishi, Yamazaki, PRC76, 045201 (2007) • Ikeda, Sato, PRC76, 035203 (2007) • Shevchenko, Gal, Mares, PRC76, 044004 (2007) • Doté, Hyodo, Weise, PRC79, 014003 (2009) • Ikeda, Kamano, Sato, PTP124, 533 (2010) • Barnea, Gal, Liverts, PLB712, 132 (2012) Resonant state of K bar NN- πΣN - πΛN coupled channel three-body system Resonance & Channel coupling ⇒ “Fully coupled -channel Complex Scaling Method”

S. Aoyama, T. Myo, K. Kato, K. Ikeda, PTP116, 1 (2006) Complex Scaling Method T. Myo, Y. Kikuchi, H. Masui, K. Kato, PPNP79, 1 (2014) … Powerful tool for resonance study of many -body system

Full ccCSM with a pheno. potential NN: Av18 pot. K bar N- πY: Akaishi -Yamazaki pot. (Pheno., E-indep.) Λ* pole = 28 MeV, Γ πΣ /2 = 20 MeV B KN K bar NN cont. The “K - pp” pole for AY-potential case: B KNN = 51 MeV, Γ πYN /2 = 16 MeV Λ*N cont. Scaling angle θ=30 ° πΛN cont. πΣN cont. Dimension = 6400 A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)

Comparison of typical calculations of K - pp Variational coupled-channel Faddeev-AGS method (Ikeda-Kamano-Sato, CSM (Dote-Hyodo-Weise, Shevchenko-Gal-Mares) Akaishi-Yamazaki, (Dote-Inoue-Myo) Barnea-Gal-Liverts) × ○ ○ Resonance Bound state Pole on complex approximation energy plane △ ○ ○ Coupled-channel Single channel calc. Explicit treatment incorporating πY channels of all channels into the effective K bar N potential × ○ ○ Wave function （△？） × ○ Potential type ○ Separable type

Hamiltonian 3 1 Meson      Baryon 2 H M T V V    NN ( MB ) ( MB )         bar Baryon K N ,      V     V    3, i ... symmetric for baryon’s site ( i=1,2)       MB MB MB MB  i 1,2 Glöckle, Miyagawa, Few-body Systems 30, 241 (2001)  NN potential = Av18 potential R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, PRC 51, 38 (1995)  K bar N- πY potential = Chiral SU(3)-based potential Theoretical energy-dependent potential A. Dote, T. Inoue, T. Myo, NPA 912, 66 (2013)  ( I 0,1) C   1       1           2   ij ( I 0,1)  : Gaussian form g r r d V r g r 3 ex p     ij ij 3 /2 ij i j i j d 2 8 f m m ij  i j  Ignore YN and πN potentials

Wave function  Baryon-Baryon are antisymmetrized on space, spin and isospin as well as label (flavor). M 3 Glöckle, Miyagawa, Few-body Systems 30, 241 (2001)  Spatial part = Correlated Gaussian function B 1 (3) x 2  including 3 types of Jacobi coordinates  projected onto a parity eigenstate of B 1 B 2 , (3) x 1 B 2

Just diagonalize the complex-scaled Hamiltonian matrix! K bar NN K bar NN K bar NN - πΛN - πΣN θ - K bar NN H ij πΣN πΣN ＊ - πΣN - πΛN πΛN ＊ ＊ - πΛN No channel elimination!!

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

Chiral SU(3)-based K bar N potential • Anti-kaon, Pion = Nambu-Goldstone boson ... governed by chiral dynamics  Coupled-channel chiral dynamics Non-relativistic potential (Chiral Unitary model)  ( I 0,1) C   1     N. Kaiser, P.B. Siegel, W. Weise, NPA 594 (1995) 325       ij ( I 0,1) V r g r E. Oset, A. Ramos, NPA 635 (1998) 99 ij i j i j 2 8 f m m  i j  Weinberg-Tomozawa term   1     2   g r 3 ex p r d of effective chiral Lagrangian : Gaussian form      ij 3 /2 ij d ij ω i : meson energy  Based on Chiral SU(3) theory → Energy dependence A. Dote, T. Inoue, T. Myo, NPA 912, 66 (2013) Constrained by K bar N scattering length • Old data: a KN(I=0) = -1.70 + i0.67 fm, a KN(I=1) = 0.37 + i0.60 fm A. D. Martin, NPB179, 33(1979) • SIDDHARTA K - p data with a coupled-channel chiral dynamics: a K-p = -0.65 + i0.81 fm, a K-n = 0.57 + i0.72 fm M. Bazzi et al., NPA 881, 88 (2012) Y. Ikeda, T. Hyodo, W. Weise, NPA 881, 98 (2012)

A. Dote, T. Inoue, T. Myo, arXiv: 1710.07589 2. “K - pp” investigated with Fully coupled-channel Complex Scaling Method • Formalism • Chiral SU(3)-based potential • Self-consistency for energy-dependent potential in coupled-channel case

How to deal with E-dep. potential? Chiral SU(3)-based potential = Energy-dependent potential  I ( 0,1) C   1           ij ( I 0,1) V r g r ij i j i j 2 8 f m m  i j • How to treat energy-dependent potentials in many-body system? • Moreover, coupled-channels case??? “Self -consistency for Meson- Baryon energy”

Outline: Self-consistent calc. for E-dep. potential  Find a pole of the K bar NN- πYN three-body system with ccCSM         H E E    M B " K p p " " K pp " " K pp "  Self-consistency for complex Meson-Baryon energy • E(MB) In : assumed in the MB potential • E(MB) Cal : calculated with the obtained “ K - pp ” E(MB) In = E(MB) Cal  Estimation of Meson- Baryon pair energy in “K - pp” • Use averaged meson binding energy B(M) An interacting MB pair carries A. D., T. Inoue, T. Myo, arXiv: 1710.07589 100% of B(M) = “Field picture” • Examine extreme two ansatz 50% of B(M) = “Particle picture” A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)

A. Dote, T. Inoue, T. Myo, arXiv: 1710.07589 3. Result

Realization of self-consistency Chiral SU(3) pot. Indicator of self-consistency (f π =110 MeV, Martin) Field picture Δ=| E(MB) Cal – E(MB) In | Indicator Δ [MeV] Self-consistent solution : B K-pp = 23.5 MeV, Γ πYN /2 = 9.1 MeV - Re E(MB) In [MeV]

Result Martin constraint SIDDHARTA constraint f π =120 MeV f π =120 MeV 120 120 100 90 90 100 (-B KNN , - Γ/2) [MeV] (-B KNN , - Γ/2) [MeV] Field: ( B, Γ/2) = ( 19 － 36, 8 － 14) Field: ( B, Γ/2) = ( 14 － 28, 8 － 15) Particle: (B, Γ/2) = (30 － 47, 12 － 14) Particle: (B, Γ/2) = (22 － 38, 13 － 18)

Results of “K - pp” J-PARC E27 DISTO J-PARC E15-1 st Faddeev-AGS Pheno. pot. (E-indep.) FINUDA Variational (Gauss) Pheno. pot. (E-indep.) Variational (Gauss) Full ccCSM (Gauss) Chial. pot. (E-dep.) Pheno. pot. (E-indep.) Faddeev-AGS Chiral pot. (E-dep.) Full ccCSM (Gauss) Chiral pot. (E-dep.) K - pp binding energy [MeV] πΣN threshold K bar NN threshold

4. Discussion K - P P

1. Dense matter or not? Chiral SU(3) potential Pheno. potential (E-dep.) (E-indep.) A. Dote, T. Inoue and T. Myo, Y. Akaishi and T. Yamazaki, NPA 912, 66 (2013) PRC 65, 044005 (2002) K - B. E. (Λ*) = 28 MeV P B. E. (Λ*) ~ 15 MeV ⇒ Λ* = Λ(1405) ⇒ Λ* = Λ(1420) B. E. (“K - pp”) = 14-38 MeV B. E. (“K - pp”) = 51 MeV K - P P NN distance = 2.2 fm NN distance = 1.9 fm ⇒ ~ 1.6 ρ 0 ⇒ ~ ρ 0 (=0.17 fm -3 )