Ds (2317) and D (2308) as candidates for tetraquarks?€¦ · string), 4-body (“flip-flop”)...

V. Dmitrasinovic, Nagoya, Dec. 08

Ds+(2317) and D*(2308) as

candidates for tetraquarks?

ヴェリコドミトラシノビッチ

Veljko DmitrašinovićLaboratory of Physics (010),Vinča Institute of NuclearSciences, Belgrade, Serbia

“New Hadrons with Various Flavors” Workshop Nagoya , 6-7 December 2008


• Introduction: scalar D mesons• ‘t Hooft interaction in the

constituent quark model (CQM) Ann.Phys.321, 355 (06)

• Charm tetraquark spectrum in CQM: D+(2308) - Ds

+(2317) Mass PuzzlePRD70,096011 (04); PRL94,162002 (05); MPLA21, 533 (06)

• Summary and Outlook

Outline:


Charmed strange mesons• Predicted spectrum in

potential models (pre-2006) :

• Two (?) old strange scalar (JP = 0+) states: Ds

**+(2317) (BaBar) and Ds

**+(2632) (SELEX, unconfirmed!?);

• New X(2690) (BaBar) state, equivalent to newDs1

**+(2700) (Belle) JP = 1- state?

• Too light ! Too many?!

317)

DsJ*(2317)

DsJ(2460)

Ds

Ds*

Ds1(2536)

Ds2(2573)

consistent with models

not consistent

to be found


Charmed nonstrange mesons• Spectrum predicted in

potential models:• Scalar (0+) nonstrange state:

D+(2308) (Belle), equivalent toD+(2405) (FOCUS) ? (2- σapart, see [I. Bigi and Orsay group, hep-ph/0512270]

• Strange (2317) and nonstrange(2308) states degenerate!?!

• The 2308 is too light. If 2405 is a different state, then too many and too heavy!


Two approaches to tetraquarks

• Colorless: “hadronic molecules”/ “mesonia” in quasi-“bootstrap” models based on Bethe-Salpeter eqn (v. Beveren & Rupp, Valencia group, Kolomeitsev & Lutz, Sassen & Krewald)

• Colorful: constituent quarks, need a model of confinement (2-body (“F.F”), 3-body (Y-string), 4-body (“flip-flop”) forces)


“True” tetraquarks • Check if D+(2308), Ds

+(2317) = (q2q*2) might be “true tetraquarks”?

• Colour quark force dependence important:Multiquarks related to multiple colour singlets (expected since 1976)

• “F.F”/“Coulomb gauge” Lorentz vector two-body confinement model leads to a q2q*2 in the colourstate

341233


The constituent quark model

• Schrödinger eqn for wave fns.• Must define the constituent-quark dynamics, • OGE + linear scalar two-body confinement, widely used

in quarkonium spectroscopy leads to unstable two-meson systems and unstable baryons!

• need vector two-body confinement, or must add as yet unknown three-quark forces (PRD67, 114007 (’03)) to the scalar two-body confinement to stabilize the nucleon & two meson states.

• Spin-dep. “hyperfine” forces, treated perturbatively. Calculate tetraquark mass!


Example of a bound two-meson state due to two-body vector colour potential

• Tetraquarks bound for large enough q/q* mass ratio. [Ader, J.M. Richard and TaxilPRD25,2370(‘82)]

• Double-b heavy-light tetraquark mass as a function of separation.(D. Janc, 2004)

• The “hidden-colour”state (88) is confined.Saturation visible in the (11) 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.


SU(6) contents of single-charm tetraquarks

• Hyperfine interactionsdetermine the spin-flavoursplittings of mass spectra1) ‘t Hooft h.f. Interaction2) colour-spin h.f. interaction

• Scalar states are generally the lowest-lying tetraquarks

� w(w w )l , s t

6 120,84,

6 120,


Hyperfine interactions in the CQM

• The “color-magnetic” CS (Fermi-Breit) interaction believed to be the main source of SU(6) multipletsplittings in CQM.

• CS HFI cannot solve the “UA(1) problem” in mesons (and lesser problems in baryons).

• The ‘t Hooft interaction solves the UA(1) problem, and improves baryon spectra.

• “Calibrate” HFI in mesons and baryons, then use it in multiquarks Ann.Phys.321, 355 (06)


Instanton-induced ‘t Hooft interaction in QCD

• Instantons induce a new three-quark, flavour-dependent, contact interaction

• Closing one pair of “legs” leads to a two-quark HF interaction that depends on flavour and spin.

uL

dR

sR

� dL

sL

uR

NF=3

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I q2L

< qq3R 3L>

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Light mesons with ‘t Hooft interaction

• Vector and pseudoscalars• Only p.s. mesons affected

by ‘t Hooft HFI• Vector mesons still ideally

mixed (contrary to Glozman-Riska int.)

• Correct ordering of states: U(1) problem solved even in non-relativistic approx.

• Only the pion (still) too heavy (non-relativistic).Ann.Phys.321, 355 (06)

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eV)

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Single-charm tetraquark SU(3) classification

• Tetraquark SU(3) C.G. series: two3*-, 6-, 15*-plets. V.D. PRD70,096011 (2004)

• The two 3*-plets and the “inner” triplet in the 15*-plet may mix!

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�

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Y YY

I3

→

3 3 3 3 3 6 15A S× × = + + +


Single-charm tetraquark masseswith ‘t Hooft h.f. interaction

• Flavour dependence of ‘t Hooft interaction breaks degeneracy.

• SU(3) symmetry breaking s-u/d quark mass difference adds to the splitting.PRD70,096011 (2004)

2.25

2.75

mas

s(G

eV)

6

K = 0 K > 0

D+K

DS+h

Expt

DS+p

2.50

3.00

S3

15

A3

m ms u,d¹

D (2632)S

D (2317)S


and mixings with ‘tHooft HFI• CG series predicts three Ds and

four D mesons• The lowest mass Ds and D in the

antisymmetric 3*-plet• Symmetric 3*-plet mixes with the

15*-plet• This mixing splits the two Ds states

into a “light” and a “heavy” one. • Antisymmetric 3*-plet D mixes with

the 6-plet and allows the small deviation of D from Ds mass to be calculated

mas

s(G

eV)

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

D+K

DS+h

D (2632)S

D (2317)S

A( )GeV

153 −S 63 −A


D+(2308) - Ds+(2317) Mass Puzzle I

• All members of the antisymmetric 3*-plet and 6-plet are degenerate irrespective of their strangeness!PRL94,162002 (2005)

• Hidden strangeness in flavour w.f.

3

3

1 ( ( ) ( ) )21 ( ( ) ( ) )2

n A

s A

D c s u s s u d d u u d

D c u u s s u d d s s d

= − − −

= − − −


D+(2308) - Ds+(2317) Mass Puzzle II

• Explains the near degeneracy of D+(2308) and Ds

+(2317). PRL94,162002 (2005)

• This mass pattern is due to the SU(3) flavour wave function permutation antisymmetry.

• Such multiplets exist only in multiquarks: the first and the simplest are found in tetraquarks!

• Similar to a0(1450) – K0*(1430) near-degeneracy in

light scalars.• Can calculate the mass difference based on the mixings

of multiplets


Summary and outlook• One antitriplet: [Ds

+(2317) (BaBar), D+(2308)(Belle)] mightbe tetraquarks lowered in mass by the ‘t Hooft interaction.

• Small strange-nonstrange meson mass difference Ds+(2317) -

D+(2308) could be a “smoking gun” evidence of tetraquarkstructure!

• Another antitriplet: [D+(2405) (FOCUS), Ds+(2632)

(SELEX), or Ds+(2690) (BaBar)] might be a mixture of cq

bar and tetraquark• The Ds

+(2632) partial decays indicate unusual flavourmultiplet (15-plet) (Zhu et al) that exists only in tetraquarks.

• Models predict many exotic tetraquarks. Experimental search is made more difficult by their large (“fall-apart”) widths!

• Many theoretical uncertainties


Exotic tetraquark mass predictions with ‘tHooft HFI

2317 2317 - - - -2724 2724 2724 2724 - -2632 2561 2520 2520 2657 23833437 3224 - - - -

3A

+sJD +

JD

6

3S

0*JsD ++

sJD +ssJD ++

JD

15

•Many exotic states, most of them within exptl reach of Belle and BaBar

•Model independent searches for exotica at present accelerators suggested by Cheng&Hou, PLB566, 193 (‘03). Two exp. search so far (CDF, BaBar).


2. Color-spin HFI in QCD

• Color-magnetic HFI from Fermi-Breitcontact in OGE.

• Its spin-dependence splits the vector and pseudoscalar mesons


Color-spin HFI in tetraquarks

• Terasaki assumes CS HFI; that leads to an isosinglet, and an isotriplet Ds

+(2317)! predicts degenerate iso-partners.

• ‘t Hooft HFI also leads to isotriplets, and many other exotics, but at higher masses.

• Model independent searches for exotica at present accelerators suggested by Cheng&Hou, PLB566, 193 (‘03). Only one exp. search so far (CDF).


mixing with CS HFI• Predict three D_s mesons: two “light” and a “heavy” one

• Lowest mass D is the ideal mixture of the antisymmetric3*-plet and the 6-plet. Lowest Ds is pure 3*-plet.

• Symmetric 3*-plet Ds,D mix with the 15*-plet ideally

• Mixing splits the two states into a heavy (hidden strangeness) and a light one

153 −S


Single-charm tetraquark masses with colour-spin h.f. interaction

• CS HFI implies no flavourdependent splitting among multiplets

• Many degenerate iso-partners due to ideal mixing!

• Two states at 2320 MeV: the iso-singlet Ds

+(2317),and iso-triplet Ds

++(2317)! • Isotriplet expected to be as

wide as the D+(2308): G = 250-350 MeV

C = 0 C 0�

m ms u,d�

(2690)

(3-15)+(2600)

(2490)

(2560)

(2350)

(2460)

(3-15)-

(2230)

(2360)

(3-6)-

(3-6)+

(2320)(2320)

(2270)


Exotic tetraquark mass predictionswith CS HFI

2320 2360 - - - -2320 2230 2250 2320 - -2460 2350 2500 2460 2550 23502690 2600 - - - -

3A

+sJD +

JD

6

3S

0*JsD ++

sJD +ssJD ++

JD

15

•Many exotic states, most of them within exptl reach of Belle and BaBar

•Model independent searches for exotica at present accelerators suggested by Cheng&Hou, PLB566, 193 (‘03). Two exp. searches so far (CDF, BaBar).


Searches for isotriplet DsJ mesons

• CDF has conducted a search for isopartnersof Ds

+(2317) (M. Shapiro) eConfC030603:MAR06,2003

• without success: no visible structure in the isotriplet spectra (lower figures)

]2

Cand. [GeV/c-π+Mass of D

2 2.2 2.4 2.6 2.8 3 3.2

Num

ber

of E

ntrie

s / 1

0 M

eV

0

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4000

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12000

-π+D

+π+D

)22.0 MeV/c± (PDG=2458.9 *02D

0π-01D 0π-*0

2D

Candidates2*0 300 D±9100

2 1.0(stat) MeV/c±Mass: 2464.2

2 2.2(stat) MeV/c±2278.1

2 4.3(stat) MeV/c±2323.4

CDF Run II Preliminary-1 ~80 pb

]2

Cand. [GeV/c-πs+Mass of D

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ber

of E

ntrie

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0 M

eV0

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-πs+D

+πs+D

CDF Run II Preliminary-1 ~80 pb) > 350 MeV/cπ(tp


DsJ’s - Overtly Exotic ?• BaBar conducted two

searches for isopartners of Ds

+(2317) so far (2004 and 2006, A. Palani)

• No clear signs of exotic charge states !

• These exotics should be very wide (G = 250-350 MeV) due to “fall-apart”nature of the decay, however.

Ds+π-

Ds+π+

DsJ(2460) ?DsJ(2317) ?

232 fb-1 hep-ex/0604030


Decays as tetraquark signature?

• Two kinds of decays measured so far (still incomplete): hadronicand EM.

• Ds+(2317) is very narrow due isospin conservation: main hadronic

decay (Ds+(1969) π0) violates isospin.

• Other predicted tetraquarks should be wider due to their being above the allowed decay thresholds.

• Ds+(2632) partial decays indicate unusual flavour multiplet (15-

plet?) that exists only in tetraquarks.Liu et al.PRD70, 094009,2004

• decays in agreement with tetraquark assumption. • Most EM decays of Ds

+(2317) consistent with cs content, however.ssB D D⎯⎯→ +


Outlook: theoretical uncertainties in multiquark calculations

• Colour confining dynamics• Hyperfine interactions• Relativity• Chiral symmetry• Dynamical pair production


Colour in Multiquarks

• Color interactions must confine quarks. • Confinement must not act between quarks in two separate

color singlets (“color saturation”), except perhaps for the v.d.Waals force.

• There are multiple color singlets and the interactions mix them. Color singlet eigenstates are mixtures that depend on the spatial separation.

• All color singlets ought to be stable, as all of them may be admixed. All color eigenstates ought to be observable: new (“hidden color”) hadrons.


Modelling Confinement• Two-body interaction

• with the “saturating” colour factor has been used.

• “Saturation” of color forces = no confining forces between colour singlets (modulo v.d.Waals force)

• Lorentz vector confining int.: all c. singlets stable.• Lorentz scalar confining int.: some c. singlets unstable!

)()()(41)(

8,..,1ijjiij

a

aj

aiij rvFFrvrV ⋅≡= ∑

=

λλ

)( ji FF ⋅


Tetraquark colour states• Color quark interactions mix the

two colour singlets• One state turns into two mesons at

asymptotic distances; another is a confined hidden-colour (hc) tetraquark state.

• Diagonal states are orthogonal: h.c. state cannot decay into two mesons.

• No physical process in the NRCQM can turn one c. eigenstateinto another. Need annihilation processes (relativity).

341234122413

341234122413

66cos33sin88

66sin33cos11

θθ

θθ

+−=

+=

0 1 2 3 4 5

0

10

20

30

40

50

Colour singlet mixing angle in tetraquarks


• A “saturating” three-body force has been introduced, V.D. PLB499,136 (2001)

• 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 a weak 3-body.

• PRD 67, 114007 (03).

( )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

Three-body force: stability and saturation


Three-quark colour force in tetraquarks

• Saturation: mixes states in the asymptotic basis, but changes only thehidden-colour stateenergy

• Does not mix states in the Pauli basis, but changes theirenergies. Unstable.PRD 67, 114007 (03).

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


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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)

Ds (2317) and D (2308) as candidates for tetraquarks?€¦ · string), 4-body (“flip-flop”)...

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