Ghil-Seok Yang, Yongseok Oh, Hyun-Chul Kim NTG (Nuclear Theory Group), (Nuclear Theory Group), Inha University Inha University HEP (Center for High Energy Physics), Kyungpook Nat‘l University Kyungpook Nat‘l University

Prehistory of SU(3) Baryons Prehistory of SU(3) Baryons Motivation (Θ +, N *) Motivation (Θ +, N *) Chiral Soliton Model Chiral Soliton Model Masses and Decay Width Masses and Decay Width Summary Summary

Naïve Quark Model Naïve Quark Model (up, down, strange light quarks): SU(3) scheme to classify particles with the same spin and parity Fundamental Particles ? SU(2) SU(3) multiplets (proton, neutron) : isospin [ SU(2) ] → higher symmetry (Σ, K,···) : SU(3) Hadron [ baryon (qqq), meson (qq) ] : SU(3) color singlet Why not 4, 5, 6, … quark states ? representation 10* (10) Nothing prevents such states to exist Y. s. Oh and H. c. Kim, Phys. Rev. D 70, (2004)

Θ , Diakonov, Petrov, and Polyakov : Narrow 5-quark resonance (q 4 q : Θ + ) ( M = 1530, Γ ~ 15 MeV from Chiral Soliton Model ) ( uddss ) T3T3 1 Θ + Θ + ( uudds ) ½-½ 2 Ξ+Ξ+Ξ+Ξ+3/2 Ξ0Ξ0Ξ0Ξ03/2 Ξ-Ξ-Ξ-Ξ-3/2 Ξ --3/2 Σ-Σ-Σ-Σ-10 Σ0Σ0Σ0Σ010 Σ+Σ+Σ+Σ+10 ( uudss ) p * p * ( uud ) n * ( udd ) n * Y S = 1 S = 0 Anti-decuplet Anti-decuplet (10) S = -1 S = -2

Successful searches for Θ + (2003~2005) : 2007 PDG Successful searches for Θ + (2003~2005) : 2007 PDG

Unsuccessful searches for Θ + (2006~2008) : 2010 PDG Unsuccessful searches for Θ + (2006~2008) : 2010 PDG ??? ?

Experimental Status Experimental Status New positive experiments ( ) Θ + ■ DIANA 2010 ( Θ + ) : M = 1538±2, Γ= 0.39±0.10 MeV (K + n → K 0 p, higher statistical significance : 6σ - 8σ) LEPS, SVD, KEK [Signals are confirmed by LEPS, SVD, KEK, …] ■ GRAAL (N* ) : M = 1685±0.012 MeV, CBELSA/TAPS, LNS-Sendai ( CBELSA/TAPS, LNS-Sendai, …) (uddss) T3T3 1 Θ + Θ + ( uudds ) ½-½ 2 Ξ+Ξ+Ξ+Ξ+ 3/2 Ξ0Ξ0Ξ0Ξ0 3/2 Ξ-Ξ-Ξ-Ξ- 3/2 Ξ -- 3/2 Σ-Σ-Σ-Σ- 10 Σ0Σ0Σ0Σ0 10 Σ+Σ+Σ+Σ+ 10 (uudss) p * p * ( uud ) n * ( udd ) n * Y S = 1 S = 0 Anti-decuplet Anti-decuplet (10) Various experimental data for Θ + and Various experimental data for Θ + and N* Mass of Θ + : 1525 – 1565 MeV ■ Mass of Θ + : 1525 – 1565 MeV Mass of : 1665 – 1695 MeV ■ Mass of N* : 1665 – 1695 MeV

: Effective and relativistic low energy theory : Large N c limit : meson field → soliton : Quantizing SU(3) rotated-meson fields → Collective Hamiltonian, model baryon states Chiral Soliton Model Hedgehog Ansatz : Collective quantization SU(2) Witten imbedding into SU(3): SU(2) X U(1)

Model baryon state Constraint for the collective quantization : Mixings of baryon states

Mixing coefficients

Octet (8) Octet (8) : J p = 1/2 + Decuplet Decuplet (10) : J p = 3/2 + Y T3T3 Y Y T3T3 1 N NN N Ξ ΞΞ Ξ Λ Σ0Σ0Σ0Σ Δ ΔΔ Δ Σ*Σ*Σ*Σ* Ξ*Ξ*Ξ*Ξ* Ω-Ω-Ω-Ω- -½ ½ Mass -½½ -3/ Mass SU(3) flavor symmetry breaking Collective Hamiltonian for flavor symmetry breakings

Two advantages offered by the model-independent approach in the χSM by the model-independent approach in the χSM. model-parameters 1. the very same set of dynamical model-parameters allows us to calculate the physical observables of all SU(3) baryons regardless of different SU(3) flavor representations of baryons, namely octet, decuplet, antidecuplet, and so on. model-parameters 2. these dynamical model-parameters can be adjusted to the experimental data of the baryon octet which are well established with high precisions. Mass : α, β, γ (for ) Mass : α, β, γ (for octet, decuplet, antidecuplet,…) Vector transitions : w i (i=1,2,…,6) Axial transitions : a i (i=1,2,…,6) [10], [10] Baryons l = l 0 (1 + c ΔT) : linear expansion coefficient of a wire, c [8] model-parameters

D.P.PE.K.PχQSM Considered Effects H SU(3) H. Input Masses [MeV] N * (1710 ?) Θ + Θ + (1539±2) Ξ -- Ξ -- (1862±2 ?) Σ πN [MeV] 4573Predicted → 41 Results I 2 [ fm ] m s α [MeV] m s β [MeV] m s γ [MeV] c Γ Θ+ [MeV] 15 for sym11.1 for sym0.71 for sym Polyakov, D.P.P : Diakonov, Petrov, Polyakov, Z. Physics. A. 359, (1997) Praszalowicz E.K.P : Ellis, Karliner, Praszalowicz, JHEP. 0405, 002 (2004) H.-Ch. Kim, K. Goeke χQSM : Tim Ledwig, H.-Ch. Kim, K. Goeke, Phys. Rev. D. 78, & Nucl. Phys. A Problems in the previous solitonic approaches Problems in the previous solitonic approaches

Octet (8) Octet (8) : J p = 1/2 + Decuplet Decuplet (10) : J p = 3/2 + Y T3T3 Y Y T3T3 1 N NN N Ξ ΞΞ Ξ Λ Σ0Σ0Σ0Σ Δ ΔΔ Δ Σ*Σ*Σ*Σ* Ξ*Ξ*Ξ*Ξ* Ω-Ω-Ω-Ω- n ( udd ) n p p ( uud ) Ξ - ( dss)Ξ - Ξ 0 Ξ 0 ( uss ) Σ-Σ-Σ-Σ- Σ+Σ+Σ+Σ+ Λ Σ0Σ0Σ0Σ0 -½ ½ Mass Δ - ( ddd )Δ - Δ ++ Δ ++ ( uuu ) Δ0Δ0Δ0Δ0 Δ+Δ+Δ+Δ+ Ω - Ω - ( sss ) Ξ*-Ξ*-Ξ*-Ξ*- Ξ*0Ξ*0Ξ*0Ξ*0 Σ*-Σ*-Σ*-Σ*- Σ*0Σ*0Σ*0Σ*0 Σ*+Σ*+Σ*+Σ*+ -½½ -3/ Mass SU(3) flavor symmetry breaking + Isospin symmetry breaking Collective Hamiltonian for flavor symmetry breakings +

D.P.PE.K.PχQSM Considered Effects H SU(3) H. Input Masses [MeV] N * (1710 ?) Θ + Θ + (1539±2) Ξ -- Ξ -- (1862±2 ?) Σ πN [MeV] 4573Predicted → 41 Results I 2 [ fm ] m s α [MeV] m s β [MeV] m s γ [MeV] c Γ Θ+ [MeV] 15 for sym11.1 for sym0.71 for sym Polyakov, D.P.P : Diakonov, Petrov, Polyakov, Z. Physics. A. 359, (1997) Praszalowicz E.K.P : Ellis, Karliner, Praszalowicz, JHEP. 0405, 002 (2004) H.-Ch. Kim, K. Goeke χQSM : Tim Ledwig, H.-Ch. Kim, K. Goeke, Phys. Rev. D. 78, & Nucl. Phys. A Problems in the previous solitonic approaches Problems in the previous solitonic approaches In order to determine the values of model parameters, “Model-independent approach” needs more information (at least, 2 inputs for antidecuplet baryons).

Mass splittings within a Chiral Soliton Model Mass splittings within a Chiral Soliton Model Formulae for Baryon Octet Masses hadronic mass part in terms of δ 1 and δ 2

Formulae for Baryon Decuplet Masses hadronic mass part in terms of δ 1 and δ 2

Formulae for Baryon Anti-Decuplet Masses hadronic mass part in terms of δ 3

D.P.PE.K.PχQSM Considered Effects H SU(3) H. Input Masses [MeV] N * (1710 ?) Θ + Θ + (1539±2) Ξ -- Ξ -- (1862±2 ?) Σ πN [MeV] 4573Predicted → 41 Results I 2 [ fm ] m s α [MeV] m s β [MeV] m s γ [MeV] c Γ Θ+ [MeV] 15 for sym11.1 for sym0.71 for sym Polyakov, D.P.P : Diakonov, Petrov, Polyakov, Z. Physics. A. 359, (1997) Praszalowicz E.K.P : Ellis, Karliner, Praszalowicz, JHEP. 0405, 002 (2004) H.-Ch. Kim, K. Goeke χQSM : Tim Ledwig, H.-Ch. Kim, K. Goeke, Phys. Rev. D. 78, & Nucl. Phys. A Problems in the previous solitonic approaches Problems in the previous solitonic approaches

★ In order to take fully into account the masses of the baryon octet as input, it is inevitable to consider the breakdown of isospin symmetry. ★ Two sources for the isospin symmetry breaking 1. mass differences of up and down quarks (hadronic part) 2.Electromagnetic interactions (EM part)

Δ M B = M B 1 – M B 2 = ( Δ M B ) H + ( Δ M B ) EM B(p) k p p p - k EM mass corrections Electromagnetic (EM ) self-energy EM [MeV]Exp. (p – n) EM 0.76±0.30 ΣΣ (Σ + – Σ - ) EM -0.17±0.30 ΞΞ (Ξ 0 –Ξ - ) EM -0.86±0.30 ( p – n ) exp ~ – MeV ( p – n ) EM ~0.76 MeV n ( udd ) n p p ( uud ) T3T3 Ξ - ( dss)Ξ - Ξ 0 Ξ 0 ( uss ) Σ-Σ-Σ-Σ- Σ+Σ+Σ+Σ+ Λ Σ0Σ0Σ0Σ0 -½ 1 ½ 1 Y Gasser, Leutwyler, Phys.Rep 87, 77 “Quark Masses”

In the ChSM, It can be further reduced to Because of Bose symmetry G. S. Yang, H.-Ch. Kim and M. V. Polyakov, Phys. Lett. B 695, 214 (2011)

Weinberg-Treiman formula M EM (T 3 ) = αT βT 3 + γ Dashen ansatz ΔM EM ~ κT 3 2 ~ κ’Q 2

Coleman-Glashow Coleman-Glashow relation EM [MeV] Exp. Exp. [input] (M p – M n ) EM0.76±0.30 (M Σ+ – M Σ - ) EM-0.17±0.30 (M Ξ 0 –M Ξ - ) EM-0.86±0.30

EM [MeV] Exp. Exp. [input]reproduced (M p – M n ) EM0.76± ±0.22 (M Σ+ – M Σ - ) EM-0.17± ±0.23 (M Ξ 0 –M Ξ - ) EM-0.86± ±0.28 Coleman-Glashow Coleman-Glashow relation Χ 2 fit

[ D.W.Thomas et al.] [ PDG, 2010 ] [ GW, 2006 ] [ Gatchina, 1981 ] Physical mass differences of baryon decuplet ■ Physical mass differences of baryon decuplet

Mass splittings within a Chiral Soliton Model Mass splittings within a Chiral Soliton Model Formulae for Baryon Octet Masses (ΔM) EM (ΔM) H hadronic mass part in terms of δ 1 and δ 2 G. S. Yang, H.-Ch. Kim and M. V. Polyakov, Phys. Lett. B 695, 214 (2011)

D.P.PE.K.PχQSM This Work Considered Effects H SU(3) H. EMHH EM + iso H. + SU(3) H. Input Masses [MeV] N * (1710 ?) Θ + Θ + (1539±2) Ξ -- Ξ -- (1862±2 ?) Θ + Θ + : MeV Σ πN [MeV] 4573Predicted → 41 Result s I 2 [ fm ] m s α [MeV] m s β [MeV] m s γ [MeV] c Γ Θ+ [MeV] 15 for sym 11.1 for sym0.71 for sym Polyakov, D.P.P : Diakonov, Petrov, Polyakov, Z. Physics. A. 359, (1997) Praszalowicz E.K.P : Ellis, Karliner, Praszalowicz, JHEP. 0405, 002 (2004) H.-Ch. Kim, K. Goeke χQSM : Tim Ledwig, H.-Ch. Kim, K. Goeke, Phys. Rev. D. 78, & Nucl. Phys. A Problems in the previous solitonic approaches Problems in the previous solitonic approaches (uddss) T3T3 1 Θ + Θ + ( uudds ) ½-½ 2 Ξ+Ξ+Ξ+Ξ+ 3/2 Ξ0Ξ0Ξ0Ξ0 3/2 Ξ-Ξ-Ξ-Ξ- 3/2 Ξ -- 3/2 Σ-Σ-Σ-Σ- 10 Σ0Σ0Σ0Σ0 10 Σ+Σ+Σ+Σ+ 10 (uudss) p * p * ( uud ) n * ( udd ) n * Y S = 1 S = 0 Anti-decuplet Anti-decuplet (10) Various experimental data for Θ + and Various experimental data for Θ + and N* Mass of Θ + : 1525 – 1565 MeV ■ Mass of Θ + : 1525 – 1565 MeV Mass of : 1665 – 1695 MeV ■ Mass of N* : 1665 – 1695 MeV

Axial-vector transitions with The full expression for the axial-vector transitions g 1 BB’ = g 1 BB’ (0) + g 1 BB’ (op) + g 1 BB’ (wf)

Axial-vector transitions 0.36±0.08

Baryon octet masses

Baryon decuplet masses

Various experimental data for Θ + and Various experimental data for Θ + and N* Mass of Θ + : 1525 – 1565 MeV ■ Mass of Θ + : 1525 – 1565 MeV Mass of : 1665 – 1695 MeV ■ Mass of N* : 1665 – 1695 MeV DIANALEPS

Ξ -- 3/2 = 1862 MeV NA49 : Mass of Ξ -- 3/2 = 1862 MeV

DIANALEPS GRAAL,SAID MAMI

DIANA LEPSDIANA?

Chiral Soliton Model Chiral Soliton Model : “model-independent approach” ● Mass splittings : SU(3) and isospin symmetry breakings with EM in the range of M Θ+ = MeV used as input ● Masses of octet and decuplet are not sensitive to the M Θ+ input. → very good agreement with experimental data pion-nucleon sigma term ● Small value of pion-nucleon sigma term is estimated. (Σ πN = MeV) ● M Θ+ = 1524 MeV [LEPS], M N* = 1685 MeV [GRAAL], Γ Θ+ = 0.38±0.11 MeV [DIANA] : reliable values within a chiral soliton model.

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