Application of the complex scaling method to hadronic resonances Λ(1405) and K - pp resonances

Transcription

1 Application of the complex scaling method to hadronic resonances Λ(1405) and K - pp resonances Akinobu Doté (KEK Theory Center, IPS / J-PARC branch) Takashi Inoue (ihon university) Takayuki Myo (Osaka Institute of Technology) 1. Introduction 2. K bar potential and Λ(1405) 3. Channel elimination by Feshbach method with coupled-channel Complex Scaling Method 4. Result of K - pp calculated with cccsm+feshbach method 5. Other quantum number case J π =1 - S =1 6. Summary and future plan Workshop on Hadrons in uclear medium II, 25. Oct. KEK Tokai campus, Tokai, Ibaraki, Japan

2 1. Introduction

3 Proton K - Λ(1405) Building block of kaonic nuclei ot a genuine 3-quark state Simple quark model can t reproduce the mass Well described with a meson-baryon molecule picture Success of chiral unitary model K - pp P K - P Prototype of kaonic nuclei The simplest kaonic nucleus Doorway to dense matter Chiral symmetry restoration in dense matter Interesting structure eutron star A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, (2004) uclear many-body system with K - 3 HeK -, pppk -, 4 HeK -, pppnk -,, 8 BeK -,

4 Current situation of K - pp study Theory Dote-Hyodo- Weise Akaishi- Yamazaki Barnea-Gal- Liverts Ikeda-Sato Shevchenko- Gal-Mares B(K - pp) 20± ~ ~ 70 Width Γ 40 ~ ~ ~ 110 Method Potential Variational (Gauss) Chiral (E-dep.) Variational (Gauss) Pheno. Variational (HH) Chiral (E-dep.) Faddeev-AGS Chiral (E-indep.) Faddeev-AGS Pheno. Kinematics on-rel. on-rel. on-rel. Rel. on-rel. Experiments B(K - pp) = 116 MeV Γ = 67 MeV if it is K - pp FIUDA B(K - pp) = 103 MeV Γ = 118 MeV if it is K - pp DISTO J-PARC E15 (Preliminary) J-PARC E27 (Preliminary)

5 Current situation of K - pp study Theory Dote-Hyodo- Weise Akaishi- Yamazaki Barnea-Gal- Liverts Ikeda-Sato Shevchenko- Gal-Mares B(K - pp) 20± ~ ~ 70 Width Γ 40 ~ ~ ~ 110 Method Potential Variational (Gauss) Variational (Gauss) Variational (HH) Faddeev-AGS Chiral Pheno. Chiral Chiral K - pp exists above π-σ- threshold! (E-dep.) (E-dep.) (E-indep.) Faddeev-AGS Pheno. Kinematics on-rel. on-rel. on-rel. Rel. on-rel. Experiments From theoretical viewpoint, (100 MeV below K bar -- threshold) K - pp = Resonance state of a K bar -πy coupled system K bar + + K bar π + Σ + B(K - pp) = 116MeV Γ = 67 MeV if it is K - pp FIUDA 1. Consider a coupled-channel problem 2. Treat resonant states adequately 3. Get the wave function to analyze the state 4. Confirmed that CSM works well on many-body systems B(K - pp) = 103MeV Γ = 118 MeV if it is K - pp coupled-channel Complex Scaling Method DISTO J-PARC E15 (Preliminary) J-PARC E27 (Preliminary)

6 Complex Scaling Method for Resonance Complex rotation of coordinate (Complex scaling) i :, U e e i r r k k H 1 U HU, U Wave functions of resonant states are transformed to be square-integrable. Diagonalizing H θ with Gaussian basis function, we can obtain resonant states, in the same way as bound states! S. Aoyama, T. Myo, K. Kato and K. Ikeda, PTP116, 1 (2006) J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269. E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280 Continuum state appears on 2θ line. Resonance pole is off from 2θ line, and independent of θ. (ABC theorem )

7 2. K bar potential and Λ(1405)

8 Phenomenological potential Chiral SU(3) potential K bar (-πy) potential Y. Akaishi and T. Yamazaki, PRC65, (2002) Energy-independent potential Constrained by the K bar scattering length and Λ(1405) energy Anti-kaon is a ambu-goldstone boson which is governed by chiral dynamics. Chiral unitary model (Oset-Ramos, Kaiser-Siegel-Weise, Jido, Hyodo, ) Weinberg-Tomozawa term, r-space, Gaussian form Semi-rela. / on-rela. Based on Chiral SU(3) theory Energy dependence A non-relativistic potential (Rv2) ( I 0,1) C ( I 0,1) 1 Vij r i j gij r 8 f m m ij 2 g ij 1 d r ex p 3/ 2 3 rdij ij i j 2 Constrained by K bar scattering length a K(I=0) = i0.67fm, a K(I=1) = 0.37+i0.60fm A. D. Martin, PB179, 33(1979) A. D., T. Inoue, T. Myo, PA 912, 66 (2013)

9 K bar system with c.c. Complex Scaling Method K bar + Λ(1405) 1435 Scattering problem π + Σ 1332 [MeV] L=0 K bar (J π =0 -, T=1/2) L=0 π (J π =0 -, T=1) (J π =1/2 +, T=1/2) Y (J π =1/2 +, T=0,1) K bar -πy coupled system with s-wave and isospin-0,1 state

10 Λ* resonance obtained with cccsm 1. Complex-scale the Hamiltonian: r re iθ, p pe -iθ 2. Diagonalize the complex-scaled H θ with Gaussian base 2θ Λ* Higher pole 2θ M [MeV] I=0 K bar -πσ system -Γ/2 [MeV] A. D., T. Inoue, T. Myo, ucl. Phys. A 912, 66 (2013) πσ continuum K bar continuum Lower pole A. D., T. Myo, ucl. Phys. A 930, 86 (2014) K bar potential: a chiral SU(3) potential (Rv2, f π =110) πσ continuum K bar continuum Double-pole structure is confirmed.

11 3. Channel elimination by Feshbach projection with cccsm πσ, πλ Q K bar P Reduce the coupled-channel problem to a single channel problem

12 Formalism of cccsm + Feshbach method Elimination of channels by Feshbash method Schrödinger eq. in model space P and out of model space Q Schrödinger eq. in P-space : P Eff P T U E E P TP vp VPQ P P E V T v QP Q Q Q Q P Effective potential for P-space U E v V G E V Eff P P PQ Q QP Q-space Green function: G Q E 1 E H QQ Extended Closure Relation in Complex Scaling Method H QQ n n n H U H U QQ QQ 1 n n 1 1 C R B Express the G Q (E) with Gaussian base using ECR Diagonalize H θ QQ with Gaussian base, n n n Well approximated T. Myo, A. Ohnishi and K. Kato, PTP99, 801 (1998) G E 1 1 E H E Q n n QQ n n n : expanded with Gaussian base. 1 G E U E v V U U Eff P P PQ QP GQ Q E V

13 4. Result of K - pp calculated with cccsm + Feshbach method

14 Apply cccsm + Feshbach method to K - pp K - pp K bar - πσ - πλ (J π =0 -, T=1/2) For the two-body system, P = K bar, Q = πy bar V K Y; I 0,1 V Y Y ' ; I 0,1 Schrödinger eq. for K bar channel : Eff Feshbach + cccsm U bar E K ( I 0,1) T V U E E Eff bar bar bar bar bar K K i ( I ) K K K i1,2 Trial wave function ( K,1) ( K,1) (3) (3) ( K " K pp" C,,1) (3), (3) a Ga x x Ga x x S 0 K T 1/ 2 a x, x x, x 0 C ( K,2) ( K,2) (3) (3) ( K,2) (3) (3) a Ga 2 Ga 2 S K T 1/ 2 a Ch. 1: K bar, : 1 E Ch. 2: K bar, : 1 O Basis function = Correlated Gaussian including 3-types Jacobi-coordinates G A x (3) ( K, i) (3) (3) ( K, i) (3) (3) ( K, i) 1 a x1, x2 a exp x1, x2 a (3) x2

15 Self-consistency for complex K bar energy K bar E K Eff U K bar I E bar i ( ) K How to determine the two-body energy in the three-body system? 1. Kaon s binding energy: B K H H H A. D., T. Hyodo, W. Weise, PRC79, (2009) : Hamiltonian of two nucleons 2. Define a K bar -bond energy in two ways E K M 2 M mk B K M mk B K : Field picture : Particle picture

16 Result pot. : Av18 (Central) K bar pot. : Rv2c potential (f π =110MeV) Fix the K bar energy at Λ* self-consistent for Λ* in free space K bar -- E K [MeV] K - pp (-28.6, -21.6) MeV Λ* (-17, -18) MeV K bar -- continuum Λ*- continuum -Γ/2 [MeV] θ = 30 deg.

17 Result K bar energy self-consistent in K - pp Field picture K bar -- pot. : Av18 (Central) K bar pot. : Rv2c potential (f π =110MeV) E K [MeV] K - pp (-25.6, -11.6) MeV K bar i 0.2 fm i 0.2 fm -Γ/2 [MeV] Λ*- continuum θ = 30 deg. K bar -- continuum

18 Result K bar energy self-consistent in K - pp Particle picture K bar -- pot. : Av18 (Central) K bar pot. : Rv2c potential (f π =110MeV) E K [MeV] K - pp (-27.3, -18.9) MeV K bar i 0.3 fm i 0.2 fm -Γ/2 [MeV] Λ*- continuum θ = 30 deg. K bar -- continuum

19 Result f π =90~120MeV / Field picture or Particle picture pot. : Av18 (Central) K bar pot. : Rv2c potential (f π =90-120MeV) Unstable for scaling angle θ! f π = 90 f π = 90 Field pict. : (B, Γ/2) = (21~32, 9~16) Particle pict. : (B, Γ/2) = (25~30, 15~32)

20 correlation density Correlation density in Complex Scaling Method x r x,, r X, r i X e x x X, pot. : Av18 (Central) K bar pot. : Rv2c potential f π =110, Particle pict. e 3i 3 2 i d R xe, R repulsive core Im ρ Re ρ K bar

21 K bar correlation density pot. : Av18 (Central) K bar pot. : Rv2c potential f π =110, Particle pict. K bar (I=0) [Re] K bar Λ* [Re] Λ* I=0 I=0,1 K bar K bar (I=0) [Im] (I=1) Λ* [Im] I=0 K bar compacter than I=1 one Strong K bar attraction in I=0 I=0 K bar seems similar to Λ* Λ* survives in K - pp

22 5. Other quantum-number case J π =1 - state S =1

23 J-PARC E27 experiment K - pp search by d (π +, K + ) reaction at 1.69 GeV/c 2-proton-coincidence analysis Inclusive spectrum: Ichikawa et al., PTEP 101D03 (2014) Ichikawa s talk at EXA2104 If the observed state is really the K - pp and if spin-flip is not so strong in the reaction, spin of K - pp formed from deuteron should be 1 because the deuteron has the spin 1. (Prof. T. Harada)

24 How is J π = 1 - state S =1? " K pp" L 0, : s wave S 0 K 1 T 1/ 2 K J π =0 -, T=1/2 T 1/ 2 " K d " LK 0, : s wave K 0 K - d studied simply with S = 1 J π =1 -, T=1/2 K - + Deuteron -like channel potential: Av4 ( 3 E, 3 O) (fitted with 5-range Gaussian functions) Tensor force is incorporated into central potential. K bar potential: A phenomenological potential Energy independent Y. Akaishi and T. Yamazaki, PRC 52 (2002) Λ*-fixed ansatz E(K bar ) in effective K bar potential is fixed to the Λ* energy.

25 How is J π = 1 - state S =1? pot. : Av4 (5 Gauss) K bar pot. : AY potential K bar threshold Λ* fixed E K [MeV] (-28, -21) MeV Λ* resonance ~ (-2, 0 ) MeV - : 4.3 fm K bar -[] : 12 fm Deuteron with distant K bar K bar -- continuum K bar -d continuum Λ*- continuum -Γ/2 [MeV] o resonance of K bar (S =1)

26 6. Summary and future plans

27 6. Summary and future plans A prototype of K bar nuclei K - pp = Resonance state of K bar -πy coupled system coupled-channel Complex Scaling Method + Feshbach projection Represent the Q-space Green function with the Extended Complete Set well approximated by Gaussian base Eliminate πy channels to reduce the problem to a K bar single channel problem. K - pp studied with cccsm+feshbch method Used a Chiral SU(3)-based potential (Gaussian form in r-space) Self-consistency for kaon s complex energy Rv2c potential case (B, Γ/2) = (21~31, 9~16) MeV : Field picture (25~30, 15~32) MeV : Particle pict. Correlation density in CSM shows Mean distance ~ 2.2 fm ormal density effect of repulsive core and Λ* survival in K - pp resonance. J π =1 - state ( Deuteron+K - -like channel) seems not to exist as a resonance state. Future plans Full-coupled channel calculation of K - pp Consider with the K bar potential constrained by the latest data of SIDDAHRTA Application to resonances of other hadronic systems