Mass spectrum of the light scalar tetraquark nonet with Glozman-Riska hyperfine interaction V. Borka  Jovanović 1  and S. ...
Outline <ul><li>light scalar tetraquarks (nonet) </li></ul><ul><li>Young diagrams for q i q j  pairs </li></ul><ul><li>Cal...
<ul><li>quarks: (u, d)  (c, s)  (t, b) + all their antiparticles </li></ul><ul><li>some quantum numbers of light quarks:  ...
<ul><li>There is a group of operators (eight in number) which do not change the interaction when they operate on it. </li>...
Scalar  tetraquarks <ul><li>they are composed of the three light flavors  u ,  d ,  s </li></ul><ul><li>total spin of this...
81 tetraquark states <ul><li>There are 81 different tetraquarks  composed of the three light flavors  u ,  d ,  s </li></u...
Spin for  ,  and  pairs of  nonet <ul><li>Scalar tetraquarks have total spin S = 0 </li></ul><ul><li>For symmetric spin wa...
For tetraquarks  ( q  =  u ,  d ,  s ),  we calculate product  λ i λ j  for combinations of two quarks: for i , j  = 1,2,3...
( 2 )   quark and antiquark may belong to 8 or 1. member of 1 : members of 8:
( 3 )   two antiquarks may belong to 3 or members of 3 : members of
Table 1.   Part of the flavor wave function of the tetraquark nonet, for certain quark combination. When we compare parts ...
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 , ...
<ul><li>The interaction we use is given by the Hamiltonian operator: </li></ul><ul><li>(L. Ya. Glozman and D. O. Riska 199...
GR HFI contribution ν -  flavor wave function - mass operator  total GR HFI contribution to masses
wave functions masses
(1) The theoretical masses of constituent quarks m u , m s  m c  and of the constant C χ  are calculated from  χ 2  fittin...
χ 2  fit We minimized the following quantity:
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  ...
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  ...
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  · (...
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...
Table 3.   The constituent quark masses m u  (= m d ) ,  m s ,   m c   , HFI constant   C χ   and the corresponding  χ 2  ...
Table 3.   The constituent quark masses m u  (= m d ) ,  m s ,   m c   , HFI constant   C χ   and the corresponding  χ 2  ...
Table 4.
<ul><li>We have made a systematic analysis of the charm tetraquark </li></ul><ul><li>states  </li></ul><ul><li>GR HFI sign...
References [1] R. L. Jaffe,  Phys. Rev. D  15 , 267 (1977) [2] T. V. Brito, F. S. Navarra, M. Nielsen, M. E. Bracco,  Phys...
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V. B. Jovanovic/ S. Ignjatovic: Mass Spectrum of the Light Scalar Tetraquark Nonet with Glozman-Riska Hyperfine Interaction

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

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V. B. Jovanovic/ S. Ignjatovic: Mass Spectrum of the Light Scalar Tetraquark Nonet with Glozman-Riska Hyperfine Interaction

  1. 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
  2. 2. Outline <ul><li>light scalar tetraquarks (nonet) </li></ul><ul><li>Young diagrams for q i q j pairs </li></ul><ul><li>Calculating of spin and flavor </li></ul><ul><li>Glozman-Riska hyperfine interaction (flavor-spin HFI) </li></ul><ul><li>Method : theoretical hadron masses, fitting mass equations, calculating constitutive quark masses </li></ul><ul><li>Conclusions </li></ul>
  3. 3. <ul><li>quarks: (u, d) (c, s) (t, b) + all their antiparticles </li></ul><ul><li>some quantum numbers of light quarks: u , d, s . </li></ul><ul><li>B - baryon number </li></ul><ul><li>T - magnitude of isospin </li></ul><ul><li>T 3 - 3-component of </li></ul><ul><li>isospin </li></ul><ul><li>σ - spin in units of ћ </li></ul><ul><li>S - strangeness </li></ul><ul><li>Y - hypercharge </li></ul><ul><li>Q - charge in units of e </li></ul>quark B T T 3 σ S Y Q u 1/3 1/2 1/2 1/2 0 1/3 2/3 d 1/3 1/2 -1/2 1/2 0 1/3 -1/3 s 1/3 0 0 1/2 -1 -2/3 -1/3
  4. 4. <ul><li>There is a group of operators (eight in number) which do not change the interaction when they operate on it. </li></ul><ul><ul><li>the SU(3) indicates that the basis of the group consists of 3 independent states (the 3 quarks) </li></ul></ul><ul><ul><li>every operator which operates in a space which is specified by 3 basis states can be written as a linear combination of these 8, augmented by the identity operator </li></ul></ul><ul><li>SU(3) provides the foundation for grouping the hadrons into supermultiplets, as the octets, decimets and singlets are collectively called. </li></ul>SU(3) operators
  5. 5. Scalar tetraquarks <ul><li>they are composed of the three light flavors u , d , s </li></ul><ul><li>total spin of this system is 0 </li></ul><ul><li>Q = T 3 + Y/2; Y = B + S + C </li></ul><ul><li>  </li></ul><ul><li>B = 1 /3 for quark , -1 /3 for antiquark </li></ul><ul><li>S = -1 for s quark , 1 for s -antiquark </li></ul><ul><li>C = 1 for c quark , -1 for c -antiquark </li></ul><ul><li>For tetraquarks with two light quarks attached to two light antiquarks, we have: </li></ul><ul><li>B = 1/3 + 1/3 – 1/3 – 1/3 = 0 ; C = 0 => Y = S </li></ul>
  6. 6. 81 tetraquark states <ul><li>There are 81 different tetraquarks composed of the three light flavors u , d , s </li></ul><ul><li>in the flavor SU(3) group, product gives the following multiplets: </li></ul><ul><li>There are two octets and they have mixed symmetry which is the permutation symmetry just of the first quark pair. Due to mutual orthogonality, one octet is mixed symmetric and the other one is mixed antisymmetric. </li></ul><ul><li>we discuss nonet, which consists of one singlet and one octet. These nine states are: </li></ul><ul><li>σ (500), f 0 (980), κ + (800), κ 0 (800), κ 0 (800), κ – (800), a 0 + (980), a 0 0 (980), a 0 – (980) </li></ul>
  7. 7. Spin for , and pairs of nonet <ul><li>Scalar tetraquarks have total spin S = 0 </li></ul><ul><li>For symmetric spin wave function, it applies S 12 = 1 and S 34 = 1, so for product of Pauli spin matrices it follows: </li></ul><ul><li>For antisymmetric spin wave function, it applies S 12 = 0 and S 34 = 0, so it follows: </li></ul>
  8. 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. 9. ( 2 ) quark and antiquark may belong to 8 or 1. member of 1 : members of 8:
  10. 10. ( 3 ) two antiquarks may belong to 3 or members of 3 : members of
  11. 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. 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
  13. 13. <ul><li>The interaction we use is given by the Hamiltonian operator: </li></ul><ul><li>(L. Ya. Glozman and D. O. Riska 1996, Phys. Rep. 268 , 263) </li></ul><ul><li>λ i - Gell-Mann matrices for flavor SU(3), σ i - the Pauli spin matrices, C χ - constant. </li></ul><ul><li>This schematic flavor-spin interaction between quarks and antiquarks leads to GR HFI contribution to tetraquark masses: </li></ul><ul><li>m i - the constituent quark effective masses: m u = m d ≠ m s </li></ul><ul><li>ν - flavor wave functions </li></ul><ul><li>m ν = m ν,0 + m ν, GR total quark masses </li></ul>Glozman-Riska hyperfine interaction (GR HFI)
  14. 14. GR HFI contribution ν - flavor wave function - mass operator total GR HFI contribution to masses
  15. 15. wave functions masses
  16. 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 &quot;Particle Data Group&quot; 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. 17. χ 2 fit We minimized the following quantity:
  18. 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. 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. 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. 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. 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. 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. 24. Table 4.
  25. 25. <ul><li>We have made a systematic analysis of the charm tetraquark </li></ul><ul><li>states </li></ul><ul><li>GR HFI significantly reduces the theoretical masses of the light scalar tetraquarks and brings them closer to their experimental masses. This fact confirms the conclusion from Brito et al. (2005) about the tetraquark nature of these light scalars. </li></ul><ul><li>We considered light scalar nonet as four-quark states and calculated their masses </li></ul><ul><li>Our predictions confirm the tetraquark nature of these light scalars </li></ul>Conclusions
  26. 26. References [1] R. L. Jaffe, Phys. Rev. D 15 , 267 (1977) [2] T. V. Brito, F. S. Navarra, M. Nielsen, M. E. Bracco, Phys. Lett. B 608 , 69 (2005) [3] J. Vijande, A. Valcarce, F. Fernandez, B. Silvestre-Brac, Phys. Rev. D 72 , 034025 (2005) [4] V. Borka Jovanović, J. Res. Phys. 31 , 106 (2007) [5] V. Borka Jovanovi ć , Phys. Rev. D 76 , 105011 (2007) [6] V. Borka Jovanovi ć , Fortschr. Phys. 56 , 462 (2008) [7] V. Borka Jovanovi ć , Phys. Lett. B , in preparation

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