Mass spectrum of the light scalar tetraquark nonet with Glozman-Riska hyperfine interaction V. Borka Jovanović 1 and S. R. Ignjatović 2 1 Laboratory of Physics (010), Vinča Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia 2 Department of Physics, Faculty of Science, Mladena Stojanovića 2, 78000 Banja Luka, Bosnia and Herzegovina
26
Embed
V. B. Jovanovic/ S. Ignjatovic: Mass Spectrum of the Light Scalar Tetraquark Nonet with Glozman-Riska Hyperfine Interaction
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Mass spectrum of the light scalar tetraquark nonet with Glozman-Riska
hyperfine interaction
V. Borka Jovanović1 and S. R. Ignjatović2
1Laboratory of Physics (010), Vinča Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia
2Department of Physics, Faculty of Science, Mladena Stojanovića 2, 78000 Banja Luka, Bosnia and Herzegovina
quarks: (u, d) (c, s) (t, b) + all their antiparticles
some quantum numbers of light quarks: u, d, s.
quark B T T3 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
B - baryon number T - magnitude of isospin T3 - 3-component of isospin
σ - spin in units of ћ S - strangeness Y - hypercharge Q - charge in units of e
There is a group of operators (eight in number) which do not change the interaction when they operate on it.
• the SU(3) indicates that the basis of the group consists of 3 independent states (the 3 quarks)
• 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
SU(3) provides the foundation for grouping the hadrons into supermultiplets, as the octets, decimets and singlets are collectively called.
SU(3) operators
Scalar tetraquarksqqqq
they are composed of the three light flavors u, d, s total spin of this system is 0
Q = T3 + Y/2; Y = B + S + C B =1/3 for quark, -1/3 for antiquarkS = -1 for s quark, 1 for s-antiquarkC = 1 for c quark, -1 for c-antiquark
For tetraquarks with two light quarks attached to two light antiquarks, we have:
B = 1/3 + 1/3 – 1/3 – 1/3 = 0; C = 0 => Y = S
81 tetraquark states There are 81 different tetraquarks composed of the
three light flavors u, d, s in the flavor SU(3) group, product gives the
following multiplets:
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.
we discuss nonet, which consists of one singlet and one octet. These nine states are:
Scalar tetraquarks have total spin S = 0 For symmetric spin wave function, it applies S12 = 1 and S34 =
1, so for product of Pauli spin matrices it follows:
For antisymmetric spin wave function, it applies S12 = 0 and S34 = 0, so it follows:
qqqq qq
1 2 3 4 1σ σ σ σ= =r r r r
1 3 1 4 2 3 2 4 2σ σ σ σ σ σ σ σ= = = = −r r r r r r r r
1 2 3 4q q q q
1 2 3 4 3σ σ σ σ= = −r r r r
1 3 1 4 2 3 2 4 0σ σ σ σ σ σ σ σ= = = =r r r r r r r r
For tetraquarks (q = u, d, s), we calculate product λiλj 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.
1 2q q
Wave functions and λiλj
1 2
1 2
83 3
33 3 3 64
6 63
λ λ
λ λ
= − ⊗ = + =
( ) ( ) ( )1 1 1, ,
2 2 2ud du ds sd su us− − −
3 :
( ) ( ) ( )1 1 1, , , , ,
2 2 2ud du ds sd su us uu dd ss+ + +
members of members of 6:
3
(2) quark and antiquark may belong to 8 or 1.
member of 1:
1 3 1 4 2 3 2 4, , ,q q q q q q q q
2 3
2 3
28 8
33 3 1 816
1 13
λ λ
λ λ
= ⊗ = + = −
( )1
3uu dd ss+ +
( ) ( ) ( ) ( ) ( )1 1 1 1 1, , , , 2
2 2 2 2 6ud du ds sd su us uu dd uu dd ss+ + + − + −
( ) ( ) ( )1 1 1, ,
2 2 2du ud sd ds us su+ + +
members of 8:
(3) two antiquarks may belong to 3 or
6.3 4q q
members of 3:
members of 6 :
( ) ( ) ( )1 1 1, ,
2 2 2ud du ds sd su us− − −
( ) ( ) ( )1 1 1, , , , ,
2 2 2ud du ds sd su us uu dd ss+ + +
3 4
3 4
83 3
33 3 3 64
6 63
λ λ
λ λ
= − ⊗ = + =
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 ofmultiplets, we can see which combination correspond to some representation. In that way λ i λ j can be calculated.
2)) · (1/2 + mu / mc) = 2520 MeV(mΩc =) 2ms + mc – (8 Cχ / (3ms
2)) · (1/2 + ms / mc) = 2698 MeV
mesonsmu = md
(MeV)
ms
(MeV)
mc
(MeV)Cχ
(10 7 MeV 3) χ2
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+, D0, D*+, D*0, Ds+, Ds*
+ 550 644 1426 4.47 5.97 x 10 – 5
ηc, J/ψ / / 1534 1.03 1.98 x 10 – 8
D+, D0, D*+, D*0, Ds+, Ds*
+,
ηc, J/ψ454 547 1524 3.96 3.51 x 10 – 4
π, Κ, η, η’,ρ, K*, ω, φ,D+, D0, D*+, D*0, Ds
+, Ds*+,
ηc, J/ψ
207 479 1624 0.527 1.062
Table 3. The constituent quark masses mu (=md), ms, mc , HFI constant Cχ and the corresponding χ2 values, obtained from fitting meson masses.
Table 3. The constituent quark masses mu (=md), ms, mc , HFI constant Cχ and the corresponding χ2 values, obtained from fitting baryon masses.
baryonsmu = md
(MeV)
ms
(MeV)
mc
(MeV)
Cχ
(10 7 MeV 3) χ2
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
Σc, Ξc+, Ξc
0, Λc, Σ*c, Ωc 658 815 1446 8.71 2.12 x 10 – 1
N, Σ, Ξ, Λ,Δ, Σ*, Ξ*, Ω,Σc, Ξc
+, Ξc0, Λc, Σ*c, Ωc
537 643 1278 0.230 1.43 x 10 – 1
Table 4.
• We have made a systematic analysis of the charm tetraquarkstates • 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.• We considered light scalar nonet as four-quark states and calculated their masses• Our predictions confirm the tetraquark nature of these light scalars
Conclusions
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 Jovanović, Phys. Rev. D 76, 105011 (2007)
[6] V. Borka Jovanović, Fortschr. Phys. 56, 462 (2008)
[7] V. Borka Jovanović, Phys. Lett. B, in preparation