Lecture form Spring School on Strings, Cosmology and Particles (SSSCP2009), March 31-4 2009, Belgrade/Nis, Serbia

1. Mass spectrum of the light scalar tetraquark nonet with Glozman-Riska hyperfine interaction V. Borka Jovanović 1 and S. R. Ignjatovi ć 2 1 Laboratory of Physics (010), Vin č a Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia 2 Department of Physics, Faculty of Science, Mladena Stojanovi ć a 2, 78000 Banja Luka, Bosnia and Herzegovina

8. For tetraquarks ( q = u , d , s ), we calculate product λ i λ j for combinations of two quarks: for i , j = 1,2,3,4 in group SU (3) F . ( 1 ) two quarks may belong to multiplet or to multiplet 6. Wave functions and λ i λ j members of members of 6 :

9. ( 2 ) quark and antiquark may belong to 8 or 1. member of 1 : members of 8:

10. ( 3 ) two antiquarks may belong to 3 or members of 3 : members of

11. Table 1. Part of the flavor wave function of the tetraquark nonet, for certain quark combination. When we compare parts of the flavor wave functions from this table with wave functions ofthe members of multiplets, we can see which combination correspond to some representation. In that way λ i λ j can be calculated.

12. Table 2. The product λ i λ j (Gell-Mann matrices for flavor SU(3)) for MS and MA multiplets: 8 MS , 8 MA and 1 MS , 1 MA . sym- metry MS λ 1 λ 2 = 4/3 λ 1 λ 3 = λ 1 λ 4 = λ 2 λ 3 = λ 2 λ 4 = 2/3 λ 3 λ 4 = 4/3 MA λ 1 λ 2 = -8/3 λ 1 λ 3 = λ 1 λ 4 = λ 2 λ 3 = λ 2 λ 4 = 2/3 λ 3 λ 4 = -8/3

14. GR HFI contribution ν - flavor wave function - mass operator total GR HFI contribution to masses

15. wave functions masses

16. (1) The theoretical masses of constituent quarks m u , m s m c and of the constant C χ are calculated from χ 2 fitting the mass equations for mesons and baryons, with GR interaction included. (2) The corresponding experimental masses are taken from "Particle Data Group" site: http://pdg.lbl.gov . (3) Tetraquark masses are calculated using values of constitutive quarks obtained from equations of meson and baryon masses. Method

17. χ 2 fit We minimized the following quantity:

18. MESONS light pseudoscalar mesons ( m π =) 2 m u – 2 C χ / m u 2 = 140 MeV (m K =) m u + m s – 2 C χ / (m u · m s ) = 494 MeV (m η =) 2m u – 2 C χ / m u 2 = 548 MeV (m η ’ =) 2m s + 16 C χ / m s 2 = 958 MeV light vector mesons (m ρ =) 2m u + 2 C χ / (3 m u 2 ) = 776 MeV (m K* =) m u + m s + 2 C χ / ( 3 m u · m s ) = 892 MeV (m ω =) 2m u + 2 C χ / (3 m u 2 ) = 783 MeV (m φ =) 2m s – 16 C χ / (3 m s 2 ) = 1020 MeV Theoretical mass equations for mesons, with GR HFI included

19. charmed mesons (m D, ± =) m u + m c – 2 C χ / (m u · m c ) = 1869 MeV (m D, 0 =) m u + m c – 2 C χ / (m u · m c ) = 1865 MeV (m D*, ± =) m u + m c + 2 C χ / ( 3 m u · m c ) = 2010 MeV (m D*, 0 =) m u + m c + 2 C χ / ( 3 m u · m c ) = 2007 MeV strange charmed mesons (m Ds, ± =) m s + m c – 2 C χ / (m s · m c ) = 1968 MeV (m Ds*, ± =) m s + m c + 2 C χ / ( 3 m s · m c ) = 2112 MeV double charmed mesons ( m η c =) 2 m c – 2 C χ / m c 2 = 2980 MeV ( m J / ψ =) 2 m c + 2 C χ / (3 m c 2 ) = 3097 MeV

20. BARYONS light baryon octet ( m N =) 3 m u – 8 C χ / m u 2 = 940 MeV ( m Σ =) 2 m u + m s – C χ / m u 2 · (1 + 7 m u / m s ) = 1190 MeV ( m Ξ =) m u + 2m s – C χ / m s 2 · (1 + 7 m s / m u ) = 1315 MeV ( m Λ =) 2m u + m s – C χ / m u 2 · ( 13 + 11 m u / m s ) /3 = 1116 MeV light baryon decuplet ( m Δ =) 3m u – 4 C χ / m u 2 = 1232 MeV ( m Σ * =) 2m u + m s – (8 C χ / (3m u 2 ) ) · (1/2 + m u / m s ) = 1385 MeV ( m Ξ * =) m u + 2m s – (8 C χ / (3m s 2 )) · (1/2 + m s / m u ) = 1530 MeV ( m Ω =) 3 m s – 4 C χ / m s 2 = 1672 MeV Theoretical mass equations for baryons, with GR HFI included

21. heavy baryons ( m Σ c =) 2 m u + m c – C χ / m u 2 · (1 + 7 m u / m c ) = 2455 MeV ( m Ξ c ,+ =) m u + 2m c – C χ / m c 2 · (1 + 7 m c / m u ) = 2470 MeV ( m Ξ c ,0 =) m u + 2m c – C χ / m c 2 · (1 + 7 m c / m u ) = 2475 MeV ( m Λ c =) 2m u + m c – C χ / m u 2 · ( 13 + 11 m u / m c ) /3 = 2285 MeV ( m Σ * c =) 2m u + m c – (8 C χ / (3m u 2 ) ) · (1/2 + m u / m c ) = 2520 MeV ( m Ω c =) 2 m s + m c – (8 C χ / (3m s 2 ) ) · (1/ 2 + m s / m c ) = 2698 MeV

22. Table 3. The constituent quark masses m u (= m d ) , m s , m c , HFI constant C χ and the corresponding χ 2 values, obtained from fitting meson masses. mesons m u = m d ( MeV ) m s ( MeV ) m c ( MeV ) C χ (10 7 MeV 3 ) χ 2 π , Κ , η , η ’ 221 451 / 0.644 7.62 x 10 – 1 ρ , K *, ω , φ 357 574 / 0.908 7.36 x 10 – 3 π , Κ , η , η ’ , ρ , K *, ω , φ 237 512 / 0.524 1.835 π , Κ , ρ , K *, ω , φ 308 487 / 2.25 1.56 x 10 – 5 D + , D 0 , D * + , D * 0 , D s + , D s * + 550 644 1426 4.47 5.97 x 10 – 5 η c , J / ψ / / 15 34 1. 03 1 . 98 x 10 – 8 D + , D 0 , D* + , D* 0 , D s + , D s * + , η c , J/ψ 454 547 1524 3.96 3.51 x 10 – 4 π, Κ, η, η’, ρ, K *, ω, φ , D + , D 0 , D* + , D* 0 , D s + , D s * + , η c , J/ ψ 207 479 1624 0.527 1.062

23. Table 3. The constituent quark masses m u (= m d ) , m s , m c , HFI constant C χ and the corresponding χ 2 values, obtained from fitting baryon masses. baryons m u = m d ( MeV ) m s ( MeV ) m c ( MeV ) C χ (10 7 MeV 3 ) χ 2 N , Σ , Ξ , Λ 436 577 / 0.847 8 . 65 x 10 – 4 Δ , Σ *, Ξ *, Ω 491 609 / 1. 43 3.87 x 10 – 6 N , Σ , Ξ , Λ , Δ , Σ *, Ξ *, Ω 427 571 / 0.575 2 . 61 x 10 – 2 Σ c , Ξ c + , Ξ c 0 , Λ c , Σ * c , Ω c 658 815 1446 8.71 2.12 x 10 – 1 N , Σ , Ξ , Λ , Δ , Σ *, Ξ *, Ω , Σ c , Ξ c + , Ξ c 0 , Λ c , Σ * c , Ω c 537 643 12 78 0.230 1.43 x 10 – 1

24. Table 4.

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