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

Outline

light scalar tetraquarks (nonet)

Young diagrams for qiqj pairs

Calculating of spin and flavor

Glozman-Riska hyperfine interaction (flavor-spin HFI)

Method: theoretical hadron masses, fitting mass equations, calculating constitutive quark masses

Conclusions

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:

σ(500), f0(980), κ+(800), κ0(800), κ0(800), κ–(800), a0+(980),

a00(980), a0

–(980)

3 3 3 3⊗ ⊗ ⊗

(15 3 6 3) 3 (27 10 8) (8 1) (10 8) (8 1)= + + + ⊗ = + + + + + + + +

1 2 3 4q q q q

Spin for , and pairsof nonet

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.

3, 6, 3, 6, 1, 8

sym-metry

MS λ1 λ2 = 4/3 λ1 λ3 = λ1 λ4 = λ2 λ3= λ2 λ4 = 2/3 λ3 λ4 = 4/3

MA λ1 λ2 = -8/3 λ1 λ3 = λ1 λ4 = λ2 λ3= λ2 λ4 = 2/3 λ3 λ4 = -8/3

i jq qi jq q i jq q

Table 2. The product λi λj (Gell-Mann matrices for flavor SU(3)) for MS and MA multiplets: 8MS, 8MA and 1MS, 1MA.

• The interaction we use is given by the Hamiltonian operator:

(L. Ya. Glozman and D. O. Riska 1996, Phys. Rep. 268, 263)

λi - Gell-Mann matrices for flavor SU(3), σi - the Pauli spin matrices, Cχ - constant.

• This schematic flavor-spin interaction between quarks and antiquarks leads to GR HFI contribution to tetraquark masses:

mi - the constituent quark effective masses: mu = md ≠ ms

ν - flavor wave functionsmν = mν,0 + mν,GR total quark masses

( ) ( ) ( )1 ;ij F F

GR i j i ji j

H Cχα λ λ σ σ

<= − −∑ r r

( ) ( )4

,2

1ij i j F F

GR i ji j i j

m Cm mν χ

α σ σν λ λ ν

< == ↑ − − ↑∑

( ) 1,1

1,ij qq

qq qqα − − = + ∨

Glozman-Riska hyperfine interaction (GR HFI)

( ), , , , ,

1 1

22MA MS GR GR MS GR MAm m mν ν νν ν ν↑ = + ⇒ = +

,ˆ

GR GRm Mν ν ν= ↑ ↑

( ) ( ) ( )1 2 3 41 2 1 3 1 3 3 4

1 2 1 3 2 3 1 4 2 4 3 4

1 1 1 1C

m m m m m m m m m m m mχσ σ σ σν λ λ σ σ λ λ λ λ ν

= − ↑ − + + + + ↑

r r r rr r

GR HFI contribution

ν - flavor wave function - mass operator ˆ

GRM

total GR HFI contribution to masses

ududσ =

0

1

2f usus dsds= +

2

14 12u

u

m m Cmσ χ= −

0 2 2

2 1 1 162 2

3f u ss s u s

m m m Cm m m mχ

= + − + +

0 0a fm m=

2

1 13 6u s

u u s

m m m Cm m mκ χ

= + − +

wave functions

masses

0a usds+ = 00

1

2a usus dsds= −

0a dsus− =

uddsκ + = 0 udusκ = 0 usudκ = dsudκ − =

(1) The theoretical masses of constituent quarks mu, ms mc

and of the constant Cχ are calculated from χ2 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

χ2 fit

We minimized the following quantity:

MESONS

light pseudoscalar mesons(mπ =) 2mu – 2 Cχ / mu

2 = 140 MeV

(mK =) mu + ms – 2 Cχ / (mu · ms) = 494 MeV

(mη =) 2mu – 2 Cχ / mu2 = 548 MeV

(mη’ =) 2ms + 16 Cχ / ms2 = 958 MeV

light vector mesons(mρ =) 2mu + 2 Cχ / (3mu

2) = 776 MeV

(mK* =) mu + ms + 2 Cχ / (3mu · ms) = 892 MeV

(mω =) 2mu + 2 Cχ / (3mu2) = 783 MeV

(mφ =) 2ms – 16 Cχ / (3ms2) = 1020 MeV

Theoretical mass equations for mesons, with GR HFI included

charmed mesons(mD, ± =) mu + mc – 2 Cχ / (mu · mc) = 1869 MeV

(mD, 0 =) mu + mc – 2 Cχ / (mu · mc) = 1865 MeV

(mD*, ± =) mu + mc + 2 Cχ / (3mu · mc) = 2010 MeV

(mD*, 0 =) mu + mc + 2 Cχ / (3mu · mc) = 2007 MeV

strange charmed mesons(mDs, ± =) ms + mc – 2 Cχ / (ms · mc) = 1968 MeV

(mDs*, ± =) ms + mc + 2 Cχ / (3ms · mc) = 2112 MeV

double charmed mesons(mηc =) 2mc – 2 Cχ / mc

2 = 2980 MeV

(mJ/ψ =) 2mc + 2 Cχ / (3mc2) = 3097 MeV

BARYONS

light baryon octet(mN =) 3mu – 8 Cχ / mu

2 = 940 MeV

(mΣ =) 2mu + ms – Cχ / mu2 · (1 + 7mu / ms) = 1190 MeV

(mΞ =) mu + 2ms – Cχ / ms2 · (1 + 7ms / mu) = 1315 MeV

(mΛ =) 2mu + ms – Cχ / mu2 · (13 + 11mu / ms) /3 = 1116 MeV

light baryon decuplet(mΔ =) 3mu – 4 Cχ / mu

2= 1232 MeV(mΣ* =) 2mu + ms – (8 Cχ / (3mu

2)) · (1/2 + mu / ms) = 1385 MeV(mΞ* =) mu + 2ms – (8 Cχ / (3ms

2)) · (1/2 + ms / mu) = 1530 MeV(mΩ =) 3ms – 4 Cχ / ms

2= 1672 MeV

Theoretical mass equations for baryons, with GR HFI included

heavy baryons(mΣc =) 2mu + mc – Cχ / mu

2 · (1 + 7mu / mc) = 2455 MeV(mΞc,+ =) mu + 2mc – Cχ / mc

2 · (1 + 7mc / mu) = 2470 MeV(mΞc,0 =) mu + 2mc – Cχ / mc

2 · (1 + 7mc / mu) = 2475 MeV(mΛc =) 2mu + mc – Cχ / mu

2 · (13 + 11mu / mc) /3 = 2285 MeV(mΣ*c=) 2mu + mc – (8 Cχ / (3mu

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

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[5] V. Borka Jovanović, Phys. Rev. D 76, 105011 (2007)

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