A Unified Model of Hadron-Hadron Interactions at Low Energies and Light Hadron Spectroscopy S. Shinmura and Ngo Thi Hong Xiem Gifu University, JAPAN Contents One-Hadron-Exchange Hadron-Hadron potentials Baryon-Baryon Interactions(HYP2015) Meson-Baryon Interactions and resonances S=-1 sector: pl-ps-k bar N interaction and L* S=-2 sector: K bar L-K bar S-pX interations and X* S=-3 sector: K bar X interaction and W * Meson-Meson Interactions and resonances S=0 sector: pp-k bar K-hh interaction and s, f 0, r, f 2 S=0 sector: ph-k bar K interaction and a 0, f S=1 sector: Kp-Kh and k, K* (S=2 sector: KK interaction) MPMB2015, Tohoku Univ., 12-14,August, 2015 Shinmura Shoji 1
Theoretical models of hadron-hadron(hh) interactions We have three typical approaches to HH interactions at low energies: First-priciple approach by LQCD Direct results of Fundamental Theory by HAL-QCD group talk by Sasaki Chiral Perturbation models Reordering of interaction diagrams based on Fundamental Symmetry Kaiser et al., Entem et al., Epellbaum et al., Haidenbauer et al. Hadron-exchange models Long-range part : hadron exchange mechanism Short-range part : short-range physics I Phenomenological Core (form factors) NSC,Julich,Our old version I Quark-model Core fss, ESC08 I LQCD-Core Our new version (Only in BB interaction) They play complementary roles Shinmura Shoji 2
Two sources: Experimental Knowledge on H-H interactions Two-body scattering NN, pn, KN, pp, Kp : Phase Shift Analyses are available LN,SN-SN,SN-LN,KN,K bar N-pL-pS: only cross section data are available. Model-independent (direct) Hypernuclear Spectroscopy Effective YN and YY interactions can be derived Final (intermediate) state interaction in hadron reactions Off-shell HH amplitudes Hadron spectroscopies provide information on HH interaction ex. L(1405) as a quasibound state of K bar N If 0 (980) as a quasibound state of K bar K Model-dependent (indirect) Shinmura Shoji 3
Hadron-Hadron Interactions at Low Energies Baryon-Baryon Interactions S= 0 NN S=-1 LN-SN S=-2 XN-LL-LS-SS S=-3 XL-XS S=-4 XX Meson-Baryon Interactions S= 1 KN S= 0 pn-hn-kl-ks S=-1 pl-ps-k bar N-hL-hS-KX S=-2 px-hx-k bar L-K bar S S=-3 K bar X Meson-Meson Interactions S= 2 KK S= 1 Kp-Kh S= 0 pp-k bar K-hp-hh S=-1 K bar p-k bar h S=-2 K bar K bar 3 bound states Coupled-Channel Problems Construction of Coupled -Channel Potentials Two-body systems Three-body systems Many-body systems Our goal is to construct a unified model describing BB, MB and MM interactions consistently Shinmura Shoji 4
One-hadron-exchange model of meson-baryon interaction Long-range part of potentials is determined by One Hadron Exchange SU(3) symmetric Interaction Lagrangian (mbb coupling constants are predetermined in BB potential model) Gaussian Form factor with a common range t-channel exchange m m B Short-range part of potentials has phenomenological strength Strengths satisfy the flavorsu(3)-symmtery Common range for all mb pairs is assumed We consider two cases of range to check the sensitivity pot I pot II r G 0.4 0.45 (fm) As a result, our potential has following form: V = (SU(3) sym. strengths) exp(-q 2 /L 2 ) + V(one-hadron-exchange potential) exp(-q 2 /L 2 ) u-channel exchange s-channel exchange m B m m B B B m B B where, L=2/r G m B Shinmura Shoji 5
The potential, the S-matrix and Residue Matrix For the s-channel exchange diagram, we introduce bare mass and bare coupling: m bare = real bare mass g i(bare) (p,e) = real bare coupling function V ij ( p, p ', E ) = g i (bare)( p, E) g j(bare) ( p ', E ) E m bare (t, + V u) ij ( p, p ', E ) Solving the L S Equation ( with relativistic kinematics), we obtain T ij ( p, p ', E ) = g i(ren )( p, E) g j(ren) ( p', E ) (t, + T u) E m ren (E ) ij ( p, p', E ) S ij (E) = i R i(e p ) R j (E p ) + S nonpole E E ij (E) p where, E p =m ren ( E p ) ( pole position), α= 1 m ren E (E p) R i (E p )= 1 4απ p on E p g i(ren) ( p on, E p ) Residue Matrix = R i (E p )R j (E p ) at E p (pole position) (For dynamical resonance, we introduce residue matrix, using S ij (dyn.pole) ) Shinmura Shoji 6
Results with our pn potential (Comparison with experimental values) pn S- and P-wave phase shifts pn : t-channel exch. s, f 0, r u-channel exch. N, D, N * (1440), S 11 (1567) s-channel exch. N, D, N * (1440), S 11 (1567) pn scattering lengths calc exp r G 0.40 0.45 S11 +0.2458 +0.2482 +0.2473±0.0043 S31-0.1496-0.1466-0.1444±0.0057 P11-0.2359-0.2340-0.2368±0.0058 P31 P13-0.1375-0.0862-0.1290-0.1316±0.0058-0.0894-0.0877±0.0058 P33 +0.6238 +0.6235 +0.6257±0.0058 fm**(2l+1) We obtain a reasonable fit to experimental data Shinmura Shoji 7
KN phase shifts Results with our KN potential (Comparison with experimental values) KN : t-channel exch. : s, f 0, a 0, r, w, f u-channel exch. : L, S (No s-channel exchange diagram) KN scattering lengths calc exp r G 0.40 0.45 S01-0.008-0.013 +0.00±0.02 S11-0.365-0.369-0.33±0.02 P01 +0.166 +0.179 +0.08±0.02 P11-0.106-0.103-0.16±0.02 P03-0.058-0.071-0.13±0.02 P13 +0.047 +0.040 +0.07±0.02 fm**(2l+1) We obtain again a reasonable fit to experimental data of KN scattering Shinmura Shoji 8
Results for K bar N scattering quantities K p threshold data: calc exp r G 0.40 0.45 g 2.35 2.36 2.36±0.04 RC 0.660 0.700 0.664±0.011 Rn 0.189 0.172 0.189±0.015 Re(a) -0.666-1.019 figure Im(a) 0.462 0.398 figure (fm) If Isospin-symmetric masses are used Re(a) -0.354-0.639 Im(a) 0.453 0.440 New parameters are only two!( ) K - p scattering length KEK SIDDHARTA DEAR r G =0.4 may be better for new result. {27} {10*} {10} {8-1}+5/9{8-2} {8-2} {1} IpN - - KN - - - - K bar N Shinmura Shoji 9
Our potentials provide L(1405) resonance as a single resonance s=1393-16i ( M=1393MeV, G=32MeV) for potential I (rg=0.40 fm) s=1406-6i ( M=1406MeV, G=12MeV) for potential II (rg=0.45 fm) Shinmura Shoji 10
We found additionl two poles! With pot I [1] s=1393-16i ( M=1393MeV, G=32MeV) [2] s=1405-130i ( M=1405MeV, G=260MeV) for pot I (r G =0.40 fm) Ratio of residue matrix elements R psps 2 : R KbNKbN 2 = 0.01:0.99 for [1] = 0.17:0.83 for [2] With pot II [1] K b N quasibound state 99% [2] a resonance mixed by K b N:pS=4:1 [1] s=1406-6i ( M=1406MeV, G=12MeV) [2] s=1395-155i ( M=1395MeV, G=310MeV) [3] s=1378-148i (M=1378MeV,G=296MeV) for pot II (r G =0.45 fm) [3] a resonance mixed by K b N:pS = 3:1 In [2][3] ps components are not small Shinmura Shoji 11
Extension to S=-2, -3 MB potentials We constructed a potential model describing simultaneously Baryon-Baryon and Meson-BaryonScattering. Based on SU(3)-symmetry and One-hadron-exchange mechanism NN, YN, YY, pn, KN, K bar N interactions at low energies, We extend the potential to S=-2 px-k bar L-K bar S-hX S=-3 K bar X and discuss existence of S-wave resonances Shinmura Shoji 12
Application to K bar -hyperon scattering Isospin Basis S=-2 and I=1/2 px + K bar L + K bar S + h X (1458) (1611) (1688) ( 1867) S=-2 and I=3/2 px + K bar S (1458) (1688) omitted S=-3 and I=0 S=-3 and I=1 K bar X K bar X (1815) (1815) Physical Mass Spectrum (Charge Basis) Shinmura Shoji 13
S=-2 and I=1/2,3/2 px + K bar L + K bar S S-wave phase shifts rg=0.40, I=1/2 rg=0.40, I=3/2 rg=0.45, I=1/2 rg=0.45, I=3/2 d33 d11 d33 d11 d44 d11 d11 d44 d44 d44 S11 S11 Shinmura Shoji 14
Resonance Poles X*(I=1/2,J p =1/2 - ) r G =0.40 I=1/2 1:pX 3:K bar L 4:K bar S s=1510-73i Cross sections s11, s33, s44 s11 s33 s44 r G =0.45 I=1/2 1:pX 3:K bar L 4:K bar S s=1495-84i s11 s33 s44 Shinmura Shoji 15
S-Wave Phase Shifts S=-3 and I=0 and 1 : K bar X scattering Isospin=0 state I=0 r G =0.45 r G =0.40 Bound state pole( Im(q)>0) for r G =0.40 at E=1796MeV(BE=19MeV) I=1 r G =0.40 r G =0.45 Virtual state pole( Im(q)<0) for r G =0.45 at E=1802MeV( BE =13MeV) Shinmura Shoji 16
The origin of the K bar -Baryon Attractions Vecotr meson exchange contributions play important roles (1)isospin-dependent r contributions (2)strongly attractive w (decreasinging with S ) (3)strondly repulsive f (increasing with S ) On-shell S-wave Potential (V/4p) at 50MeV above each K bar -Baryon threshold K bar -B Isospin r w f scalar Baryon Short Total K bar N 0-42.4-91.7 20-25.0 22.4-44.9-161.5 L*(1405) K bar N 1 14.1-91.7 20-25.3 156.5 11.5 85.2 K bar L 1/2 0-84.7 49.6-28.7 6.5-12.1-69.4 K bar S 1/2-78.4-87.5 55.9-27.1 30.7 18.5-87.9 K bar S 3/2 39.2-87.5 55.9-30.0 0-2.2-24.6 K bar X 0-69.4-70.3 90.7-28.7 14.2 7.1-56.4 } X*(1510) W*(1796) K bar X 1 23.1-70.3 90.7-32.7 5.2 14.2 30.4 Scalar mesons provide almost constant attraction (~-30MeV) Shinmura Shoji 17
Hadron-Hadron(H-H) Interactions at Low Energies Baryon-Baryon Interactions S= 0 NN S=-1 LN-SN S=-2 XN-LL-LS-SS S=-3 XL-XS S=-4 XX Meson-Baryon Interactions S= 1 KN S= 0 pn-hn-kl-ks S=-1 K bar N-pL-pS-hL-hS-KX S=-2 px-hx-k bar L-K bar S S=-3 K bar X Meson-Meson Interactions S= 2 KK S= 1 Kp-Kh S= 0 pp-k bar K-hp-hh S=-1 K bar p-k bar h S=-2 K bar K bar Coupled-Channel Problems Construction of Coupled -Channel Potentials Two-body systems Three-body systems Many-body systems Shinmura Shoji 18
One-meson-exchange model of meson-meson interactions SU(3) symmetric 3-meson interaction Lagrangian Consistent with meson-baryon(mb) potential model Ips-ps-vector (used in mb potential) L ppv =g ppv Tr [(( μ P)P P( μ P))V μ ] Ips-ps-scalar (used in mb potential) L pps =( f pps /m π )Tr [( μ P μ P) S] Ips-ps-tensor (not used in mb potential) L ppt =(2g ppt / m π )Tr[( μ P ν P)T μ ν ] Form factors : we try two types of form factors to check the sensitivity Monopole type Gaussian type Shinmura Shoji 19
Cutoff Form factors Monopole form factors For t-, u-channel exchange of vector mesons For s-channel exchange of vector mesons For s-channel exchange of scalar and tensor mesons F (q)= Λ2 +m v 2 Λ 2 +q 2 F (ω p )= Λ2 2 +m v Λ 2 2 +ω p F (ω p )= Λ4 4 +m v Λ 4 4 +ω p Gaussian form factors For t-,u-channel exchange For s-channel exchange F (q)=exp( q 2 /Λ 2 ) F (ω p )=exp( ω p 2 / Λ 2 ) Shinmura Shoji 20
S=0, Isospin=0, s- and d-wave interactions Ipp-K bar K-hh (I=0) s-wave resonances: Imonopole Igaussian exp If 0 1000-i20 1075-i170 (970-1010)-i(20-50) Is 1 580-i380 430-i380 (400-550)-i(200-350) Is 2 410-i560 390-i500 Ipp-K bar K-hh (I=0) d-wave Ratio of Residue Matrix elements at the pole (monopole) R pp 2 : R KbK 2 : R hh 2 = 0.41:0.59:0.00 for f 0 = 0.48:0.33:0.18 for s 1 = 0.98:0.017:0.002 for s 2 Ff 0,s 2 : pure dynamical Ss 1 : s-channel e-exchange (m ren (E)=E) s 1 and s 2 have much different charactor! resonance: Imonopole Igaussian exp If 2 1270-i110 1250-i90 (1275.1±1.2)-i93 M bare =1354.2MeV(monopole) Ratio of Residue Matrix elements at the pole R pp 2 : R KbK 2 : R hh 2 = 0.66:0.27:0.07 f 2 has only small hh and 27% K b K components. Shinmura Shoji 21
S=0, Isospin=0 and 1, p-wave interactions K bar K(Isospin=0) resonance: Imonopole Igaussian exp If 1016.5-i1.6 1022.5-i1.6 1019-i2.1 m bare =1150MeV E=m ren (E)=1016.5-i1.6 MeV E=m ren (E)=1022.5-i1.6 MeV Ipp-K bar K-ph(Isospin=1) resonance: Imonopole Igaussian exp Ir 800-i60 800-i60 775-i74 m bare =1220.3MeV(monopole),1047.1MeV(gaussian) Ratio of Residue Matrix elements at the pole R pp 2 : R KbK 2 : R ph 2 = 0.67:0.0005:0.33 r has no K b K component! But 1/3 comes from ph Shinmura Shoji 22
I S=0, Isospin=1, s-wave interaction ph-k bar K(I=0) Resonance pole: Imonopole Igaussian exp Ia 0 845-i15 800-i15 980-i(25-50) Trajectory of m re (E) E Im(m*) Re(m*) Moving pole: m re (980)=980-i25 S-matrix pole: m re (E)=E E=845-i15(monopole) E=800-i15(gaussian) m bare =1235.7MeV(monopole) Ratio of Residue Matrix elements at the pole R ph 2 : R KbK 2 = 0.14:0.86 Phase shifts a 0 has large K b K component. Shinmura Shoji 23
Kp-Kh, Ispspin=1/2, s- and p-wave interactions resonances: Imonopole Igaussian exp I k 1450-i75 1440-i35 (1375-1475)-i(95-175) kk 650-i230 650-i190 (653-711)-i270 Ratio of Residue Matrix elements at pole (monopole) R Kp 2 : R Kh 2 = 0.92:0.08 for k(1450) = 0.96:0.04 for k(650) resonance: Imonopole Igaussian exp IK* 907-i20 910-i18 892-i25 Kp dominant Both k mesons originate from s-channel k-exchange with m bare =1557.6MeV(monopole),1522.1MeV(gaussian) m bare =1530.8MeV(monopole),1473.3MeV(gaussian) Ratio of Residue Matrix elements R Kp 2 : R Kh 2 = 0.63:0.37 Kh component is not small (37%). Shinmura Shoji 24
Summary 1. We proposed a model of hadron-hadron interactions: Long-range part : One-hadron-exchange mechanisms Short-range part : LQCD cores (in BB potedntials) Phenomenological cores (in mb potentials) Monopole and Gaussian form factors (in mm potentials) the flavor-su(3)-symmetry assumed (but for exchanged hadron masses, we use physical masses. The SU(3) breaking comes from only this origin) gives a good description of baryon-baryon, meson-baryon, mesonmeson interactions. 2.We discussed on resonance states L*(1405) three poles are found. One is K bar K quaibound state. X*(1510, J p =1/2 - ) and W*(1796, J p =1/2 - ) are predicted. Properties and structure of meson resonances s 1, s 2, f 0, r, f 2, a 0, f, k, K* purely dynamical resonances : f 0, s 2 All other meson resonances originate from s-channel meson-exchange Shinmura Shoji 25