Physics programs (1) - KEK...Hyperfine Interactions • The spin-dependent “color-magnetic” CS...

V. Dmitrasinovic Vinca Institute of Nuclear Sciences

Tetraquarks

Veljko DmitrašinovićLaboratory of Physics (010), Vinča Institute of Nuclear Sciences, Belgrade, Serbia & Montenegro

• Experiment: new charmed mesons• Three theoretical scenarios• Quark model with tHooft force• Mesons and baryons• Colour dependent confining forces• Tetraquarks• Pentaquarks• Summary

Outline:

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) ?

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) .

• 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

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)

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

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�

Two solutions

QCD Instantons � �t Hooft�Kobayashi�Kondo�Maskawa

quark interaction�

LtH � GtH

� �� det

� �� i

� �GtHRe det� ��

GtH �f ��

��m�� m��

� � �m�K

�h�qqi��

QCD ��Nc limit � Veneziano�Witten quark interaction�

LVW � GVW

� �� det

� �� i�

� ��GVW

hIm det

� �� i�

GVW �f ��

��m�� m��

� � �m�K

�h�qqi��

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.

� dL

< qq3R 3L>

q1L q1L

< 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��

��

�� Æ��

��

� ��

�� ! "��

�� "#

�� $� %��

��

� Æ�� Æ��

��

��&

��

�� ' ��

��

� ��

• 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

Mesons

• Mesons form a nonet = singlet + octet

• For every spin and parity allowed by angular momentum conservation, P- and C-conjugation ��plet

� ��

��

I3 ��

�plet

• Confining two-body potential (simple= HO realistic= linear + Coulomb) determines overall masses (shell separation)

• Mass splitting determined by HFI

� �

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

SU(3)K = 0

�c > 0

��

S.break.

��

0.75 ��

� ��

��

� ��

�c = 0 �c = 0

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.

SU(3)C = 0� C > 0� ExptS.break.

��

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

aiij rvFFrvrV ⋅≡= ∑

)( ji FF ⋅

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 .

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.

2 2 23 0 0

1, , exp[ ( ) / ]8

a b cabcb i j k i j k i j k

V r r r d U r r r aλ λ λ

= − + +r r r

abc a b ci j kd λ λ λ

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)?

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!

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]

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

Colour singlet mixing angle in tetraquarks

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!

0 0.5 1 1.5 2 2.5 3

s [fm]

all config.triplet config.singlet config.sextet config.octet config.

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

13 246 6

5 2 /18−

5 /(9 3)

5 2 /(18 3)−

5 2 /(9 3)−

5 /(18 3)−

5 /18−

abc a b ci j kd λ λ λ

0 1 2 3 4 5 6

s [fm]

eV] Uo = 0

Uo = -20MeV (a.=3fm)Uo = -40MeV (a.=3fm)

Effects of three-quark colour force on double-charm tetraquarks (D. Janc)

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!

��

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

Charmed tetraquark mass splitting

• KKMT interaction splits the degeneracy.

• SU(3) symmetry breaking s-u/d quark mass difference adds to the splitting.

SU(3)K = 0 K > 0

D��

DS��

DS�D�

S.break.

D (2460)S

D (2320)S

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

DS��

D (2460)S

D��D (2317)S

A(GeV)

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

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).

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?

Appendix: Variational calculation (D. Janc )• Gaussian spatial configurations:

( )2 2 21 1 1 2 2 3 3R exp a x a x a x∝ − − −

Courtesy of D. Janc

• The two coupled channel Schroedinger equations are solved variationally:

• The Ansatz consists of 3 parts:• Orbital part:• Colour part:

• 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

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

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.)

0 1 2 3 4 5

s [fm]

Uo = 0 Uo = 500MeV

0 1 2 3 4 5

s [fm]

Uo = 0 Uo = 600MeV

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

0 1 2 3 4 5

s [fm]

Uo = 0 Uo = 600MeV

0 0.5 1 1.5 2 2.5 3 3.5

Vo = 0 Vo = -40MeV

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