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V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Tetraquarks
Veljko DmitrašinovićLaboratory of Physics (010), Vinča Institute of Nuclear Sciences, Belgrade, Serbia & Montenegro
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
• Experiment: new charmed mesons• Three theoretical scenarios• Quark model with tHooft force• Mesons and baryons• Colour dependent confining forces• Tetraquarks• Pentaquarks• Summary
Outline:
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Experiment
• New charmed (nonexotic) mesons: • D+(2308), Ds
+(2317) and Ds+(2632)
• Too many states:Tetraquarks?• Other candidates: X(3872) ?• Exotic hyperons: Θ+(1540) pentaquark!• Other candidates: Ξ-- (1862), Θc (3099) ?
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Three theoretical scenarios • The D+(2308), Ds
+(2317), Ds+(2632), and
X(3872) =(q2q2) “true tetraquarks”, bound by QCD forces. Colour quark force important:Multiquarks related to additional colour singlets (expected since 1976). Consistent with “true pentaquark”interpretation of Θ+(1540): Θ+ =(q4q) .
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
• The hadronic molecule model of D+(2308), Ds
+(2317), Ds+(2632), and σo(600),
resonances (I. Nakamura +V.D., many others).Consistent with Θ+(1540) as a molecular bound state of three hadrons (Kishimoto and Sato): Θ+
= K+ n πo No colored quarks, just hadrons. Chiral symmetry the only guide.
• The Θ+(1540) is a chiral soliton state; no quarks, just meson fields. What are the new mesons?
)980(),980( 00 fa
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Constituent quark model with instanton-induced interaction
• Nonrelativistic constituent quarks (massive, 330 MeV): no chiral symmetry!
• Confinement via chromo-harmonic 2-body force: a) all colour singlets stable; b) mesons and baryons asymptotic states; c) predicts new confined “hidden colour” states
• HFI determines “fine structure” of the spectrum: a) colour-spin (Fermi-Breit one-gluon exchange) b) flavour-spin (tHooft instanton induced)
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Hyperfine Interactions
• The spin-dependent “color-magnetic” CS (Fermi-Breit) interaction was believed to be the main source of SU(6) multiplet splittings in QCD.
• CS model cannot solve the “UA(1) problem” in mesons and several problems in baryons.
• The ‘t Hooft interaction solves the UA(1) problem, but it changes other mass splittings.How?
• “Calibrate” HFI in mesons and baryons, then use it in multiquarks
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
What is the �UA��� problem��
�� The sum of � and �� masses does not satisfy the Gell�
Mann�Okubo relation�
m�� �m��
� � ����� MeV�� �� �m�K � ���� MeV��
�� The � � �� mixing angle �p�s� is not the ideal one�
�p�s� � ���o �� ���o�
�
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Two solutions
QCD Instantons � �t Hooft�Kobayashi�Kondo�Maskawa
quark interaction�
LtH � GtH
hdet
� ���� � ������ det
� ����� �����i
� �GtHRe det� ���� � �����
where
GtH �f ��
���m�� �m��
� � �m�K
�h�qqi���
QCD ��Nc limit � Veneziano�Witten quark interaction�
LVW � GVW
hdet
� ���� � ������ det
� ����� �����i�
� ��GVW
hIm det
� ���� � �����i�
where
GVW �f ��
���m�� �m��
� � �m�K
�h�qqi���
�
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Instanton induced interaction in QCD
• Instantons induce a new three-body, flavour-dependent, contact interaction
• Closing one pair of “legs” leads to a two-body interaction that depends on flavour and spin.
uL
dR
sR
� dL
sL
uR
NF=3
I
uL
dR
sR
dL
sL
uR
I
�
NF=3
I q2L
< qq3R 3L>
I
q1R
q2L
q1L q1L
q2R
q1R
q2R
< qq3R 3L>
V. Dmitrasinovic Vinca Institute of Nuclear SciencesTwo new UA��� symmetry breaking interactions
There are also two �antisymmetric tensor� interactions�
�Phys� Rev� D ��� � �� ��
�t Hooft
LT� � GT�hdetf
� ������� � ������ detf
� �������� �����i
Veneziano�Witten
LT� � GT�hdetf
� ������� � ������ detf
� �������� �����i�
where ��� is the Pauli tensor
��� �i
����� ���
for two �avours this leads to an extended NJL Lagrangian
�Phys� Rev� D ��� � ������� �������
LENJL � ���i�� �m�q�� �GS�� ����� � � ��i�������
�GT �� �������� � � ��i����������
What is the physics of the Pauli tensor interaction�
�
V. Dmitrasinovic Vinca Institute of Nuclear Sciences��� ������� �� ����� ���� ���� �� �� ��
��� �������� ������ �� ������ �
��� � ������������� ��� ��� � ���� ��� � ���
����� �
���
���
�� � ���
� ���
����� ��� ��� ������� � ! "������ �� �����
��� ����� ������"#
��� $� %���� ���� ���� ��� �� �� �� ��� ������
�� �� ������ �
���� � ��������
����
�����
��� � ����
� ��� � ������ � ���
����� �
���
���&
���
�����
�� � �� ' �����������
��
� ���
�
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
• Must check our “new” model against known phenomena (see also Bonn and Tokyo groups)
• Mesons: tHooft interaction yields OK spectra in chiral models (e.g. NJL, OGE BSE), but in nonrelativistic ones?
• Baryons: Little understanding, in spite (because?) of Bonn U. papers.
Basic check: mesons and baryons
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Mesons
• Mesons form a nonet = singlet + octet
• For every spin and parity allowed by angular momentum conservation, P- and C-conjugation ��plet
I3
Y
� ����
� ����
�����
ududs
�����
Y
I3 �����
�plet
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
• Confining two-body potential (simple= HO realistic= linear + Coulomb) determines overall masses (shell separation)
• Mass splitting determined by HFI
� �
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Light meson spectrum with nonrelativistic KKMT interaction
• Vector and pseudoscalars
• Only p.s. change.• Vector mesons still
ideally mixed.• Only the pion still
too heavy.0.25
0.5
mass
(GeV
)
SU(3)K = 0
�c > 0
������
������
Expt
����
S.break.
������
1.0
0.75 ����
�
� ����
K > 0
������
��������
������
������
� ����
�
�c = 0 �c = 0
K > 0
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Comparison with Glozman-Riskainteraction
• Simple flavour-spin (Gamow-Teller) HF interaction normalized on baryons leads only to ή(960) meson OK.
• The rest is bad.• Accident? No, GR
guessed one piece of tHooft HFI correctly.
0.25
0.5
mass
(GeV
)
SU(3)C = 0� C > 0� ExptS.break.
����
0
1.0
0.75
�
�
��
������
������
����
������
������
��������
������
������
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Modelling Confinement
• Two-body interaction with the “saturating” colour factorhas been used.
• “Saturation” of color forces = no confining forces between colour singlets
• Lorentz vector confining interaction
• Nogami and Bhaduri’s or Grenoble parametrization.
)()()(41)(
8,..,1ijjiij
a
aj
aiij rvFFrvrV ⋅≡= ∑
=
λλ
)( ji FF ⋅
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Lorentz vector vs. scalar
• Lorentz scalar confining two-body potentials are not positive definite, i.e., some colour singlets are unconfined
• They lead to small spin-orbit splitting /potential
• They demand a strong three-body potential to confine the baryon!
• May require four-body forces to stabilize the tetraquark.
• Lorentz vector confining two-body potentials are stable and saturating.
• They lead to large spin-orbit splitting /potential.
• A fit to meson and baryon spectra leaves little room for a three-body force.
• May lead to bound tetraquarks for sufficiently asymmetric masses .
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Three-body force: stability and saturation
• A “saturating” three-body force has been introduced (PLB499,136)
• Two scenarios: (a) Allow all color states, demand that color 8, 10 be heavy; (b) Forbid all non-singlets
• Scenario (a) demands 3-, 4-,5-body forces etc.; (b) allows only dominant 2-body and weak 3-body.
( )3
2 2 23 0 0
1, , exp[ ( ) / ]8
b
a b cabcb i j k i j k i j k
V V V
V r r r d U r r r aλ λ λ
→ +
= − + +r r r
18
abc a b ci j kd λ λ λ
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Why tetraquarks? Scalar mesons
• Similar problems already exist in the spectrum of light scalar mesons:
• More states than fill one nonet, but not enough for two nonets.
• Supernumerary flavour singlets: glueballs? The second isovector a(1450)?
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Why tetraquarks? Charmed mesons
• Expected spectrum in potential models:
• Too many observed scalar (0+) states: strange Ds
+(2317)(Belle) and Ds
+(2632)(SELEX); and nonstrange D+(2308) (Belle)
• Too light; strange and nonstrange degenerate!
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Tetraquarks: colour states• Simplest multiquark with
two colour singlets is the tetraquark:
• In the “asymptotic” basis: are the “two-meson” and the “hidden-colour” state
• Mathematicallyequivalent (“Pauli”) basis
• “Asymptotic basis” is physically preferable;
• “Pauli” basis is more convenient (Pauli princ.)
12 34 12 341 1 , 8 8
13 24 13 243 3 , 6 6
−
q−
q[ q q]
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
More on tetraquark colour states• The quark interactions mix the
two colour singlets• One two-meson state at
asymptotic distances; another is a confined hidden-colourtetraquark state.
• Latter state cannot decay into two mesons.
• Two diagonal states areorthogonal: two separate “worlds”.
• No physical process (in NRCQM) can take one “world”into another. Need anihilation processes.
34123412 88cos11sin θ+θ
0 1 2 3 4 5
0
10
20
30
40
50
Colour singlet mixing angle in tetraquarks
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Example of a bound two-meson state due to two-body colour potential (Courtesy D. Janc )
• At short distances new state is admixed in the two-meson continuum: a resonance or bound state.
• Tetraquark is bound for large enough mass ratio:double-t heavy-light tetraquark mass as a function of separation.
• Both Pauli- and hidden -colour states confined! Saturation of the two-meson state!
10500
10550
10600
10650
10700
10750
10800
10850
10900
10950
11000
0 0.5 1 1.5 2 2.5 3
s [fm]
M [M
eV]
all config.triplet config.singlet config.sextet config.octet config.
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Three-quark colour force in tetraquarks
• The 3-body force: • Saturation: mixes
states in the asymptotic basis, but changes onlythe hidden-colour state energy
• Does not mix states in the Pauli basis, but changes theirenergies. Unstable!
12 341 1
12 341 1 12 348 8
12 348 8
13 243 3
13 243 3
13 246 6
13 246 6
5 2 /18−
5 2 /18−
5 /(9 3)
5 /(9 3)
5 2 /(18 3)−
5 2 /(18 3)−
0
0
0
5 2 /(9 3)−
5 2 /(9 3)−
5 /(18 3)−
5 /(18 3)−
5 /18−
5 / 9
5 /18
18
abc a b ci j kd λ λ λ
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
3850
3875
3900
3925
3950
3975
4000
0 1 2 3 4 5 6
s [fm]
E [M
eV] Uo = 0
Uo = -20MeV (a.=3fm)Uo = -40MeV (a.=3fm)
Effects of three-quark colour force on double-charm tetraquarks (D. Janc)
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Single-charm tetraquark SU(3) contents
• Tetraquark SU(3) C.G. series: 3-,6-, 15-plets.
• Two 3-plets and “inner” triplet in 15-plet may mix!
�
�
�
�
���
Y YY
I3
→
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Single-charm tetraquark SU(6) contents
• SU(6) multiplets: 6, 84, 120.
• Pauli principle allows only 6, and 120-plet in the ground state.
�(� � � �� � �
6, ,12084
6,120
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Charmed tetraquark mass splitting
• KKMT interaction splits the degeneracy.
• SU(3) symmetry breaking s-u/d quark mass difference adds to the splitting.
3.0
2.25
2.5
2.75
mass
(GeV
)
�
SU(3)K = 0 K > 0
D��
DS��
Expt
DS�D�
S.break.
S3
15
A3
D (2460)S
D (2320)S
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
3-15 mixing
• Only symmetric 3-plet mixes with 15-plet
• Mixing separates two states
• Low mass asymm. 3-plet
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
2.2
2.4
2.6
2.8
3
3.2
3.4
DS��
D (2460)S
D��D (2317)S
A(GeV)
mass
(GeV)
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Mixing angles
• The symmetric 3-plet mixes with the 15-plet
• Two mixing angles: strange Ds
+ and nonstrange D mesons.
• Both mixing angles small: - 5 to -10 degrees.
• Ds+(2632) partial decay
widths indicate 15-plet.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
-70
-60
-50
-40
-30
-20
-10
0
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
D+(2308) - Ds+(2317) Mass Puzzle
• Two SU(3) multiplets have unusual mass patterns due to their permutation symmetries:
• All members of the antisymmetric 3-plet and 6-plet are degenerate irrespective of their strangeness! Only in tetraquarks.
• Explains degeneracy of D+(2308) and Ds
+(2317).
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Summary of tetraquarks
• Strange resonances: Ds+(2317) (Belle) and
Ds+(2632) (SELEX); and nonstrange D+(2308)
may be tetraquarks lowered in mass due to KKMT interaction.
• Small strange-nonstrange meson mass difference Ds
+(2317) - D+(2308) is “smoking gun”evidence for tetraquarks?
• Ds+(2632) partial decay widths indicate 15-plet.
• Model predicts many exotic tetraquarks. Experiment?
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Appendix: Variational calculation (D. Janc )• Gaussian spatial configurations:
( )2 2 21 1 1 2 2 3 3R exp a x a x a x∝ − − −
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Courtesy of D. Janc
• The two coupled channel Schroedinger equations are solved variationally:
• The Ansatz consists of 3 parts:• Orbital part:• Colour part:
or
• Spin part :
• → spin 0 tetraq. ... 24 config. Ri(s1,s2,s3)CjSj
• spin 1 tetraq. ... 36 config.• optimization of parameters
si, s = a, b, c• select best configuration• build up basis (∼50 config.)
( )2 2 21 1 1 2 2 3 3R exp a x a x a x∝ − − −
iC =12 34 12 341 1 , 8 8
12 34 12 34 12 34 11 0 , 0 1 , 1 1
13 24 13 243 3 , 6 6
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Results of two-body potential variational calculation
• Only heavy-light systems may bind. Light flavour tetraquarks are resonances e.g..
• Example: Double-heavy (open bottom) double-light tetraquark
• Tbb (S = 1, I = 0, P = +)• (B.Silvestre-Brac, C.Semay (1993);
D.M.Brink, Fl.Stancu (1998); D.Janc (2003))• Threshold: • M(Tbb ) = 10523MeV • M (B + B* ) = 10651MeV →
binding energy: -128 MeV• Expansion in basis:
Ri(s1,s2; s3 = fixed )CjSj
bbud bd bu+KK
)980(),980( 00 fa
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Role of tHooft in charm-strange channel
• S=0: thresholds: D + K = 1869 + 494 = 1865 + 498 MeV
• = 2363 MeV (expt.)• = 2406 MeV (theory)• Ds + π = 1969 +140 =
2109 MeV (exp.)• = 2132 MeV (theory.)
2100
2200
2300
2400
2500
2600
2700
0 1 2 3 4 5
s [fm]
M [M
eV]
Uo = 0 Uo = 500MeV
2100
2200
2300
2400
2500
2600
2700
0 1 2 3 4 5
s [fm]
E [M
eV]
Uo = 0 Uo = 600MeV
V. Dmitrasinovic Vinca Institute of Nuclear Sciences
Role of tHooft force in charm-strangechannel
• Ds + η = 1969 + 547 = 2416 MeV
• = 2441 MeV (tHooft force Uo=600 MeV)
• S=1: thresholds: • D* + K = 2540MeV• D + K* = 2540MeV• Ds + ρ = 2773MeV• Ds* + π =2237MeV• Ds* + η = 2535MeV • (Uo =600MeV)• Silvestre-Brac, Semay:• S=0, M=2357MeV• S=1, M=2456MeV
2100
2200
2300
2400
2500
2600
2700
0 1 2 3 4 5
s [fm]
E [M
eV]
Uo = 0 Uo = 600MeV
2500
2600
2700
2800
2900
3000
3100
0 0.5 1 1.5 2 2.5 3 3.5
Vo = 0 Vo = -40MeV
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