J. Vijande

J. Vijande

EFB22/ J. Vijande 1

Multiquark systems from a quark model perspective

The 22nd European Conference on Few-Body Problems in Physics.

EFB22/ J. Vijande 2

The november revolution: 1974

BNL

SLAC

M = 3.1 GeV

0 MeV

M = 3.105 GeV

< 1.3 MeV

SLAC

M = 3.695 GeV

= 2.7 MeV

EFB22/ J. Vijande 3

EFB22/ J. Vijande 4

T. B

arne

s et

al.,

Phy

s. R

ev. D

72, 0

5402

6 (2

005)

A quiet period: 1974-2003

scnc S. Godfrey and N. Isgur, Phys. Rev. D32, 189 (1985)

cc

nc sc

EFB22/ J. Vijande 5

The beginning of a new era: 2003

Ds0*(2317), JP=0+, <3.8 MeV

Ds1(2460), JP=1+, < 3.5 MeV

X(3872)

<2.3 MeV

EFB22/ J. Vijande 6

X(3872)Z(3930) DsJ(2317)

D0(2308)

DsJ(2700) DsJ(2860)

Y(3940)

X(4160)

X(4260)

Y(4350)

Y(4660)

Z(4430)Z1(4040)

Z2(4240)DsJ(3040)

DsJ(2460)

X(4008)

X(3940)

Etc...May you live in interesting times

How to proceed?

EFB22/ J. Vijande 7

...321 qqqqqqqqqqqqmeson

...321 qqqqqqqqqqqqqqbaryon

Naive Quark Model

133

1333

166

133

3333

qqqq

qqqq

133

166

111

33333

qqqqq

qqqqq

qqqqq

1333

13631633

13361666

333333

qqqqqq

qqqqqqqqqqqq

qqqqqqqqqqqq

18881888

11881818

18811111

333333

qqqqqqqqqqqq

qqqqqqqqqqqq

qqqqqqqqqqqq

EFB22/ J. Vijande 8

Solving the Schrödinger equation for a 4q system: VM and HH

JM Color Isospin Spin R

12

3

1

2

3

4

1,2 c 3,4 n

ccnn

12

3

1

2

3

4

1,2 c 3,4 n

cncn

12 3434 12Color 3 3 , 6 6

C-parity is a good symmetry.

12 34 12 34Color 1 1 , 8 8

Pauli principle must be imposed.

Hyperspherical Harmonics: Radial part is expanded into HH functions, hyperangular part, (up to a Kmax value) and a sum of Laguerre functions, hyperradial part.

)()(,,, KnLKnR Phys. Rev. D 79, 074010 (2009). 32312123

22

21Exp

fedcba

Variational method: Radial part is expanded into generalized gaussians. Each generalized gaussian contains an infinite number of relative angular momentum, L=0 and positive parity

VMHH

0.3633861.40.3673860.6

RMSERMSE

L=0 S=1 I=0 ccnn––

EFB22/ J. Vijande 9

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

4100

4200

4300

4400E

(M

eV)

0 + +

(2 4 )0 +

(2 2 )1 + +

(2 0 )1 +

(2 2 )2 + +

(2 6 )2 +

(2 8 )1 +

(1 9 )1

(1 9 )0 +

(1 7 )0

(1 7 )2 +

(2 1 )2

(2 1 )

Energías del sistema 4q

J.V.

, et

al.,

Phy

s. R

ev. D

76, 0

9402

2 (2

007)

System: ccnn. Model: BCN

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

4100

4200

4300

4400E

(M

eV)

0 + +

(2 4 )0 +

(2 2 )1 + +

(2 0 )1 +

(2 2 )2 + +

(2 6 )2 +

(2 8 )1 +

(1 9 )1

(1 9 )0 +

(1 7 )0

(1 7 )2 +

(2 1 )2

(2 1 )

M1M2 threshold

4q energies

There are no non-exotic deeply four-quark bound sta

tes (compact)

EFB22/ J. Vijande 10

The gift from nature to hadronic physicists

nnbbnncc and

!!

These states cannot camouflage themselves in the mesonic

jungle


3800

3900

4000

4100

4200

4300

4400

4500

4600E

(MeV

)

0+

(2 8)1+

(24)2+

(30)0-

(21)1-

(21)2-

(21)0+

(28)1+

(24)2+

(30)0-

(21)1-

(21)2-

(21)

I=1I=0

4q energiesM1M2 threshold

J.V.

, A.V

., N

.B.,

Phy

s. R

ev. D

79, 0

7401

0 (2

009)

System: cncn. Model: CQC

One compact state in the ccnn sy

stem (J

P =1+ )


3800

3900

4000

4100

4200

4300

4400

4500

4600

E (

MeV

)

0 +

(2 8 )1 +

(2 4 )2 +

(3 0 )0

(2 1 )1

(2 1 )2

(2 1 )0 +

(2 8 )1 +

(2 4 )2 +

(3 0 )0

(2 1 )1

(2 1 )2

(2 1 )

I= 1I= 0

3800

3900

4000

4100

4200

4300

4400

4500

4600

E (

MeV

)

0 +

(2 8 )1 +

(2 4 )2 +

(3 0 )0

(2 1 )1

(2 1 )2

(2 1 )0 +

(2 8 )1 +

(2 4 )2 +

(3 0 )0

(2 1 )1

(2 1 )2

(2 1 )

I= 1I= 0

3800

3900

4000

4100

4200

4300

4400

4500

4600

E (

MeV

)

0 +

(2 8 )1 +

(2 4 )2 +

(3 0 )0

(2 1 )1

(2 1 )2

(2 1 )0 +

(2 8 )1 +

(2 4 )2 +

(3 0 )0

(2 1 )1

(2 1 )2

(2 1 )

I= 1I= 0

xz

y

1

2

3

4

1,2 c 3,4 n

ccnn

Bound.

Unbound.


No compact bound states in the ccnn and bbnn sectors.

One compact bound state in the ccnn sector and four/three in the bbnn sector.


What about the existence of slighty bound states very close to the threshold?


We solved the scattering of two-meson systems in a coupled-channel approach by means of the Lippmann-Schwinger equation, looking for attractive channels.

Molecular states, how to look for them?

The meson-meson interacting potential is obtained from the same quark-quark interaction used in the HH and VM methods, by means of the adiabatic approximation.

(I)

D D

cn

We study the consequences of allowing for the reordering of

quarks (I) or not (II).

D D

cn

**DDDD DD D*D*

(II)


There are no charged partners of the X(3872) [diquark-antidiquark]

JPC(I

)=1+

+(0

)

(I) DD*

(II) DD* – J/

X(3872)

T. F.-C., A

.V., J.V., Phys. R

ev. Lett. 103, 222001 (2009)

Hidden flavor sector: Charmonium


T. F. Caramés et al., Phys. Lett. B. 699, 291 (2011)(I) JP = (0) 1+

II I

Formalisms based on meson-meson and four-quark configurations are fully compatible if they incorporate all the relevants basis vectors (channels)!

Meson-M

eson

PDD*

PD*D*

PDD

Four-quark states


Hid

den

flav

orE

xpli

cit f

lavo

rThere should not be a partner of the

X(3872) in the bottom sector

There should be a JP=1+ bound state

in the exotic bottom sector

T.F.

C.,

A.V

., J.

V., P

hys.

Let

t. B

709

, 358

(20

12)

Charm Bottom

Implementing confinement

Mesons: Including the flux tube.

12rVmeson

Baryons: two-body Vs. Many-body

aaxrrrV

aarrrV

JJJJ

MBbaryon

Bbaryon

73.133)(min

5.12

3)(

2

321

1323122

a a

a

xxx

133 Color

1333 Color

J

EFB22/ J. Vijande 19/37

Tetraquarks: One step further.

231424133412

231424133412

2

8

5

4

4216

3

rrrrrr

rrrrrrrV

jiijjiBtetra

)(min 4321,

llklkklk

Steinertetra

rrrrr

V

),(min

),(min

'1'111

23142413

VV

rrrrV FFtetra

FFtetraSteinertetraMBtetra VVV ,min

166

1333333:

qqqq

qqqqColor

1

2

3

4

k l


• The Steiner piece is negligible, less than 1 % contribution, as compared to the Flip – Flop interaction

• The ground state energy of a system containing identical quarks and antiquarks is found below threshold.

• For Flavor-exotic binding increases with the mass ratio.

• For non-exotic states the effect is opposite.


0 1 2 3 4 5

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

M/m

En

ergy

dif

fere

nce

eepp ,,, eeee ,,,more stable than

2.2/ mMbecomes unstable for mmMM ,,, qqQQ

qqQQ

• The Steiner piece is negligible, less than 1 % contribution, as compared to the Flip – Flop interaction

• The ground state energy of a system containing identical quarks and antiquarks is found below threshold.

• For Flavor-exotic binding increases with the mass ratio.

• For non-exotic states the effect is opposite.

Improvements

• The effect of statistics is neglected → gluon degrees of freedom are integrated out.

• Therefore, the color structure is not included.

• Long range forces between color singlet are not discussed.

• Spin dependent terms are not taken into account.PRD76, 114013 (2007)


Pentaquarks: increasing difficulties

couplingscolor many too16

32

jiijjiBpenta rV

)(min 54321,,

mmlmlklkkmlk

Steinerpenta

rrrrrrr

V

npermutatioquark any being },,,{

,,min 1

lkji

rrrVr

V

lkjMBbaryonii

FFpenta

FFpentaSteinerpenta

MBpenta

VV

V

,min

133

166

111

33333:

qqqqq

qqqqq

qqqqq

Color

k

l

m

k

l

mj


• Models based on two-body color-additive interactions offer no bound states.

• The Steiner piece is once more negligible.

• Systems made of identical quarks and antiquarks, are found to be below the dissociation threshold when many-body configurations are considered.

• Systems containing an infinitely massive quark(antiquark) are also bound

Improvements

• Completely symmetric S-wave function has been considered. Therefore, no statistics is taken into account.

• Realistic heavy quark masses should be analyzed, paying special attention to the charm and bottom sectors, including spin-spin forces.

• Doubly-heavy systems should be addressed.

,qqqqq

.,, QqqqqqQqqq

J-M. Richard. PRC81, 015205 (2010)


Hexaquarks: reaching the limit

couplingscolor many too16

32

jiijjiBhexa rV

)

(min

65

4321,,,

6

nlnnml

mmklkknmlk

Steinerq

rrrr

rrrrr

V

npermutatioquark any being },,,,,{

,,,,min

6

nmlkji

rrrVrrrV

V

nmlMBbaryonkjiMBbaryon

FFq

FFqSteinerq

MBq

VV

V

66

6

,min

1333

13631633

13361666

333333:

qqqqqq

qqqqqqqqqqqq

qqqqqqqqqqqq

Color


)

(min

65

4321,,,

33

nlnnml

mmklkknmlk

Steinerqq

rrrr

rrrrr

V

654321 ,,,,

33

rrrVrrrV

V

MBbaryonMBbaryon

FFqq

FFqqqqq

qqqqSteinerqq

MBqq

VV

VV

V

33333

3333

33

,

,,min

2

24

nspermutatio },,{ and },,{

,,,,min

2433

onmkji

rrVrrrrV

V

okmesonnmjiSteinertetra

qqqq

nspermutatio },,{ and },,{

,,,min

333 2

onmkji

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qqq

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Color


• Q3q3 is found to be stable upto values M/m ~ 8 – 10, an intermediate region between the charm and bottom sectors.

• For baryonium, the lowest threshold corresponds to a 4q-2q configuration. However, for larger values of M/m no multiquark configuration can compete with the compact Q3 and systems become unstable with respect to the heavy baryon-

light antibaryon threshold.

Improvements

• Based on the previous results Steiner-tree diagrams are not considered.

• Antisymmetrization and color structure is once more neglected

• Annihilation is not considered.

PR

D85

, 014

019

(201

2)

33qQ

33qQ


0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

M/m

Ene

rgy

diff

eren

ce

Going beyond the adiabatic aproximation.


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PRD87, 034040 (2013)

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Summary

• Constituent quark models are an important tool to study heavy hadron spectroscopy, however, like all powerfull tools have to be handled carefully.

• Hidden flavor components, unquenching the quark model beyond the “naive” approximation, seem to be neccessary to tame the bewildering landscape of hadrons, but an amazing folklore is borning around.

• Compact four-quark bound states with non-exotic quantum numbers are hard to justify while “many-body (medium)” effects do not enter the game.

• Exotic many-quark systems should exist if our understanding of the dynamics does not hide some information. I hope experimentalists can answer this question to help in the advance of hadron spectroscopy.

• The role of antisymmetrization and understanding the adiabatic aproximation for confinement is very important to prevent a proliferation of multiquarks.


Acknowledgements

Thanks!

Let me thank the people I collaborated with in the different subjects I covered in this talk

A. Valcarce (Univ. Salamanca, Spain)

T. F. Caramés (Univ. Salamanca, Spain)

N. Barnea (Hebrew Univ. Jerusalem, Israel)

E. Weissman (Hebrew Univ. Jerusalem, Israel)

J.M. Richard (Grenoble, France)

F. Fernández (Univ. Salamanca, Spain)

H. Garcilazo (IPN, Méjico)

B. Silvestre-Brac (Grenoble, France)

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J. Vijande