Shinmura Shoji 1

A Unified Model of Hadron-Hadron Interactions

at Low Energies and Light Hadron Spectroscopy

S. Shinmura and Ngo Thi Hong Xiem Gifu University, JAPAN

MPMB2015, Tohoku Univ., 12-14,August, 2015

Contents

■One-Hadron-Exchange Hadron-Hadron potentials ■Baryon-Baryon Interactions(HYP2015)

■Meson-Baryon Interactions and resonances S=-1 sector: pL-pS-KbarN interaction and L*

S=-2 sector: KbarL-KbarS-pX interations and X* S=-3 sector: KbarX interaction and W*

■Meson-Meson Interactions and resonances S=0 sector: pp-KbarK-hh interaction and s, f0, r, f2

S=0 sector: ph-KbarK interaction and a0, f S=1 sector: Kp-Kh and k, K* (S=2 sector: KK interaction)

Shinmura Shoji 2

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 3

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,KbarN-pL-pS: only cross section data are available. 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 KbarN If0(980) as a quasibound state of KbarK

Model-dependent (indirect)

Two sources:

Model-independent (direct)

Shinmura Shoji 4

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-KbarN-hL-hS-KX S=-2 pX-hX-KbarL-KbarS S=-3 KbarX Meson-Meson Interactions S= 2 KK S= 1 Kp-Kh S= 0 pp-KbarK-hp-hh S=-1 Kbarp-Kbarh S=-2 KbarKbar

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

3 bound states

Shinmura Shoji 5

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 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 rG 0.4 0.45 (fm) As a result, our potential has following form: V = (SU(3) sym. strengths)×exp(-q2/L2) + V(one-hadron-exchange potential)×exp(-q2/L2) where, L=2/rG

m

m

m

m

m

m

m

B

B

B

B

B

B

B

B

t-channel exchange

u-channel exchange

s-channel exchange

Shinmura Shoji 6

The potential, the S-matrix and Residue Matrix

For the s-channel exchange diagram, we introduce bare mass and bare coupling:

mbare = real bare mass

gi(bare)(p,E) = real bare coupling function

Residue Matrix = Ri(Ep)Rj

(Ep) at Ep (pole position)

(For dynamical resonance, we introduce residue matrix, using Sij (dyn.pole) )

V ij( p , p ' , E ) =g i (bare )

( p , E) g j(bare)( p ' , E )

E−mbare

+ V ij

(t , u)

( 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 )

E−mren(E )+ T ij

(t , u)

( p , p ' , E )

S ij(E) = iRi(E p) R j (E p)

E−E p

+ S ij

nonpole(E)

where ,

E p=mren( E p) ( pole position) , α=√1−∂mren

∂ E(E p)

Ri(E p)=1

4απ √pon

E p

g i(ren)( pon , E p)

Shinmura Shoji 7

pN scattering lengths calc exp rG 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 -0.1375 -0.1290 -0.1316±0.0058 P13 -0.0862 -0.0894 -0.0877±0.0058 P33 +0.6238 +0.6235 +0.6257±0.0058 fm**(2L+1)

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)

Results with our pN potential

(Comparison with experimental values)

pN S- and P-wave phase shifts

We obtain a

reasonable fit

to experimental data

Shinmura Shoji 8

Results with our KN potential

(Comparison with experimental values)

KN scattering lengths calc exp rG 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)

KN phase shifts

KN : t-channel exch. : s, f

0, a

0, r, w, f

u-channel exch. : L, S (No s-channel exchange diagram)

We obtain again

a reasonable fit

to experimental data

of KN scattering

Shinmura Shoji 9

Results for Kbar N scattering quantities

K―p threshold data: calc exp rG 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)

DEAR

SIDDHARTA KEK

New parameters are only two!(●) {27} {10*} {10} {8-1}+5/9{8-2} {8-2} {1} IpN ○ ○ ○ ○ － －

KN ○ ○ － － － －

KbarN ○ ○ ○ ○ ● ●

If Isospin-symmetric masses are used

Re(a) -0.354 -0.639

Im(a) 0.453 0.440

rG=0.4 may be better for new

result.

K-p scattering length

Shinmura Shoji 10

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 11

We found additionl two poles !

[1] √s=1393-16i ( M=1393MeV, G=32MeV) [2] √s=1405-130i ( M=1405MeV, G=260MeV) for pot I (rG=0.40 fm)

[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 (rG=0.45 fm)

[3] a resonance mixed by KbN:pS = 3:1

Ratio of residue matrix elements |RpSpS|2:|RKbNKbN|2= 0.01:0.99 for [1] = 0.17:0.83 for [2]

[1] KbN quasibound state 99% [2] a resonance mixed by KbN:pS=4:1

With pot I

With pot II

In [2][3] pS components are not small

Shinmura Shoji 12

We extend the potential to

S=-2 pX-KbarL-KbarS-hX S=-3 KbarX

and discuss existence of S-wave resonances

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, KbarN interactions at low energies,

Extension to S=-2, -3 MB potentials

Shinmura Shoji 13

Application to Kbar-hyperon scattering

Physical Mass Spectrum (Charge Basis)

Isospin Basis S=-2 and I=1/2 pX + Kbar L + Kbar S + h X

(1458) (1611) (1688) ( 1867)

S=-2 and I=3/2 pX + Kbar S

(1458) (1688) S=-3 and I=0 S=-3 and I=1

Kbar X Kbar X (1815) (1815)

omitted

Shinmura Shoji 14

S=-2 and I=1/2,3/2

pX + Kbar L + Kbar S

rG=0.40, I=1/2 rG=0.40, I=3/2 rG=0.45, I=1/2 rG=0.45, I=3/2

d11 d11 d11 d11

d33

d33

d44 d44

d44 d44

S-wave phase shifts

S11 S11

Shinmura Shoji 15

Resonance Poles X*(I=1/2,Jp=1/2

－)

Cross sections s11, s33, s44

s33

s44

s11

rG=0.40

I=1/2

1:pX

3:KbarL

4:KbarS

rG=0.45

I=1/2

1:pX

3:KbarL

4:KbarS

s11

s33

s44

√s=1510-73i

√s=1495-84i

Shinmura Shoji 16

S=-3 and I=0 and 1 : Kbar X scattering

rG=0.40

rG=0.45

rG=0.40

rG=0.45

I=0

I=1

Isospin=0 state

S-Wave Phase Shifts

Bound state pole( Im(q)>0) for rG=0.40 at E=1796MeV(BE=19MeV)

Virtual state pole( Im(q)<0) for rG=0.45 at E=1802MeV(“BE”=13MeV)

Shinmura Shoji 17

The origin of the Kbar

-Baryon Attractions

Kbar-B Isospin r w f scalar Baryon Short Total

KbarN 0 -42.4 -91.7 20 -25.0 22.4 -44.9 -161.5

KbarN 1 14.1 -91.7 20 -25.3 156.5 11.5 85.2

KbarL 1/2 0 -84.7 49.6 -28.7 6.5 -12.1 -69.4

KbarS 1/2 -78.4 -87.5 55.9 -27.1 30.7 18.5 -87.9

KbarS 3/2 39.2 -87.5 55.9 -30.0 0 -2.2 -24.6

KbarX 0 -69.4 -70.3 90.7 -28.7 14.2 7.1 -56.4

KbarX 1 23.1 -70.3 90.7 -32.7 5.2 14.2 30.4

L*(1405)

X*(1510) }

W*(1796)

Scalar mesons provide almost constant attraction (～-30MeV)

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 Kbar-Baryon threshold

Shinmura Shoji 18

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 KbarN-pL-pS-hL-hS-KX S=-2 pX-hX-KbarL-KbarS S=-3 KbarX Meson-Meson Interactions S= 2 KK S= 1 Kp-Kh S= 0 pp-KbarK-hp-hh S=-1 Kbarp-Kbarh S=-2 KbarKbar

Coupled-Channel Problems Construction of Coupled -Channel Potentials Two-body systems Three-body systems Many-body systems

Shinmura Shoji 19

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) Ips-ps-scalar (used in mB potential) Ips-ps-tensor (not used in mB potential) Form factors : we try two types of form factors to check the sensitivity Monopole type Gaussian type

L ppv=g ppvTr [((∂μ P)P−P(∂

μ P))Vμ]

L pps=( f pps /mπ)Tr [(∂

μ P∂μP)S ]

L ppt=(2gppt /mπ)Tr[(∂μ P∂ν P)T

μ ν]

Shinmura Shoji 20

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

Gaussian form factors

For t-,u-channel exchange For s-channel exchange

F (q)=Λ

2+mv

2

Λ2+q

2

F (ω p)=Λ

2+mv

2

Λ2+ω p

2

F (ω p)=Λ

4+mv

4

Λ4+ω p

4

F (q)=exp(−q2/Λ

2)

F (ω p)=exp(−ω p

2/Λ

2)

Shinmura Shoji 21

S=0, Isospin=0, s- and d-wave interactions

resonances: Imonopole Igaussian exp If0 1000-i20 1075-i170 (970-1010)-i(20-50)

Is1 580-i380 430-i380 (400-550)-i(200-350)

Is2 410-i560 390-i500

resonance: Imonopole Igaussian exp If2 1270-i110 1250-i90 (1275.1±1.2)-i93

Ratio of Residue Matrix elements at the pole (monopole) |Rpp|2:|RKbK|2:|Rhh|2 = 0.41:0.59:0.00 for f0

= 0.48:0.33:0.18 for s1

= 0.98:0.017:0.002 for s2

Ff0,s2: pure dynamical Ss1 : s-channel e-exchange (m

ren(E)=E)

→s1 and s2 have much different charactor !

Mbare=1354.2MeV(monopole) Ratio of Residue Matrix elements at the pole |Rpp|2:|RKbK|2:|Rhh|2 = 0.66:0.27:0.07 → f2 has only small hh and 27% KbK components.

Ipp-Kbar

K-hh (I=0) s-wave

Ipp-Kbar

K-hh (I=0) d-wave

Shinmura Shoji 22

S=0, Isospin=0 and 1, p-wave interactions

KbarK(Isospin=0)

Ipp-KbarK-ph(Isospin=1)

resonance: Imonopole Igaussian exp If 1016.5-i1.6 1022.5-i1.6 1019-i2.1

resonance: Imonopole Igaussian exp Ir 800-i60 800-i60 775-i74

mbare=1220.3MeV(monopole),1047.1MeV(gaussian) Ratio of Residue Matrix elements at the pole |Rpp|2:|RKbK|2:|Rph|2 = 0.67:0.0005:0.33 → r has no KbK component! But 1/3 comes from ph

→ E=mren(E)=1016.5-i1.6 MeV → E=mren(E)=1022.5-i1.6 MeV

mbare=1150MeV

Shinmura Shoji 23

I S=0, Isospin=1, s-wave interaction

Resonance pole: Imonopole Igaussian exp Ia0 845-i15 800-i15 980-i(25-50)

Im(m*) Re(m*)

Moving pole: mre(980)=980-i25 S-matrix pole: mre(E)=E E=845-i15(monopole) E=800-i15(gaussian)

Trajectory of mre(E)

E

Phase shifts

mbare=1235.7MeV(monopole) Ratio of Residue Matrix elements at the pole |Rph|2:|RKbK|2 = 0.14:0.86 → a0 has large KbK component.

ph-Kbar

K(I=0)

Shinmura Shoji 24

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

resonance: Imonopole Igaussian exp IK* 907-i20 910-i18 892-i25

Ratio of Residue Matrix elements at pole (monopole) |RKp|2:|RKh|2 = 0.92:0.08 for k(1450) = 0.96:0.04 for k(650) Both k mesons originate from s-channel k-exchange with mbare =1557.6MeV(monopole),1522.1MeV(gaussian)

mbare=1530.8MeV(monopole),1473.3MeV(gaussian) Ratio of Residue Matrix elements |RKp|2:|RKh|2 = 0.63:0.37 → Kh component is not small (37%).

Kp dominant

Shinmura Shoji 25

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, meson- meson interactions. 2.We discussed on resonance states L*(1405) three poles are found. One is KbarK quaibound state. X*(1510, Jp=1/2-) and W*(1796, Jp=1/2-) are predicted. Properties and structure of meson resonances s1, s2, f0, r, f2, a0, f , k, K* purely dynamical resonances : f0, s2

All other meson resonances originate from s-channel meson-exchange