1. On the nature of the X(3872) J. Vijande University of Valencia (Spain)
A. Valcarce, T. F. Caramés (U. Salamanca)
23 November, November, 2018
On the nature of the X(3872)

2. Motivation: New charmonium (open charm) mesonsDD
3872
cc mass spectrum
Below the DD threshold charmonium spectroscopy is a good example of the simple color Fermi-Breit structure of the heavy hadron spectra. Above this threshold new experimental data indicate a more complicated situation.
X (3872), X (3940),Y (3940), Z (3940), Y(4140),...
Charmonium
Open charm
DsJ*(2317), DsJ(2460), D0*(2308),DsJ(2632), DsJ*(2700), ...
R.L. Jaffe, Phys. Rev. D15, 267 (1977)
Meson-Meson molecules, four-quark compact bound states, diquarks configurations…
23 November, November, 2018
On the nature of the X(3872)

3. Could these new mesons be four quark states?
X(3872)
X,Y,Z(3940)
Y(4260)
Z+(4430)
DD|S(0++)
DD*|S(1++)
DsDs|S(0++)
DD1|S
D*D1|S
Could these new mesons be four quark states?
Charmonium
23 November, November, 2018
On the nature of the X(3872)

4. Solving the Schrödinger equation1 2 3 1 2 3 4
1,2  c
3,4  n
ccnn 1 2 3 1 2 3 4
1,2  c
3,4  n
cncn
See Talk by A. Valcarce
On Saturday at 11:40
C-parity is a good symmetry.
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.
Phys. Rev. D 79, (2009).
23 November, November, 2018
On the nature of the X(3872)

5. Interacting potentials
Parameters determined on meson spectroscopy
BCN
Confinement: Linear potential
One-gluon exchange: Standard Fermi-Breit potential
Confinement: Linear screened potential
One-gluon exchange: Standard Fermi-Breit potential
Scale dependent as
Boson exchanges: Chiral symmetry breaking
Not active for heavy quarks
CQC
Parameters determined on the NN interaction and meson/baryon spectroscopy
23 November, November, 2018
On the nature of the X(3872)

6. J. Vijande et al., Phys. Rev. D76, 094022 (2007)
cncn (I=0). BCN model
4q Energy
Theoretical threshold
J. Vijande et al., Phys. Rev. D76, (2007)
23 November, November, 2018
On the nature of the X(3872)

7. J. Vijande et al., Phys. Rev. D79, 074010 (2009)
cncn (I=0). CQC Model
Theorerical Thresholds
No deeply bound (compact) states in the ccnn sector.
J. Vijande et al., Phys. Rev. D79, (2009)
23 November, November, 2018
On the nature of the X(3872)

8. + + n c n – c – c n – cncn w J/ c n – D — D
MULTIQUARK states , although stationary in a potential or bag, do not in general correspond to stable hadrons or even resonances. Far from it, most, perharps even all, fall apart into valence mesons and baryons without leaving more than a ripple on the meson-meson or meson-baryon scattering amplitude. If the multiquark state is unsually light or sequestered from the scattering channel, it may be prominent. If not, it is just a silly way of enumerating the states of the continuum. n c + n – c – c n –
cncn w
J/ c n – + D — D
23 November, November, 2018
On the nature of the X(3872)

9. What about the existence of slighty meson-meson bound states very close to the threshold?
23 November, November, 2018
On the nature of the X(3872)

10. Solving the Lippmann-Schwinger equation for the two meson system
(II)
23 November, November, 2018
On the nature of the X(3872)

11. (I) (II) Coupled channels DD DD* D*D* [(cn)(nc)] JPC (I) (S,L)
[(c c) (nn)] DD
0+ + (0)
(0,0)
c - 
DD*
1+ (+) (0)
(1,0),(1,2)
J/ - 
1+ (–) (1)
J/ - 
D*D*
(0,0),(2,2)
1+ – (0)
c - 
1– – (0)
(0,1),(2,3)
J/ -
1– + (0)
(1,1)
2+ + (0)
(2,0),(2,2)
2– – (0)
(2,1),(2,3)
0– + (1)
1+ – (1)
J/ - 
2+ + (1)
(I)
(II)
23 November, November, 2018
On the nature of the X(3872)

12. Interacting potentials
Parameters determined on meson spectroscopy
BCN
Confinement: Linear potential
One-gluon exchange: Standard Fermi-Breit potential
Confinement: Linear screened potential
One-gluon exchange: Standard Fermi-Breit potential
Scale dependent as
Boson exchanges: Chiral symmetry breaking
Not active for heavy quarks
CQC
Parameters determined on the NN interaction and meson/baryon spectroscopy
23 November, November, 2018
On the nature of the X(3872)

13. Non-strange two- baryon systems Strange two- baryon systems S=-2 two-
A. Valcarce et al., Rep. Prog. Phys. 68, 965 (2005)
H. Garcilazo et al., Phys. Rev. C76, (2007)
Strange two-
baryon systems
Predictions
A. Valcarce et al., Unpublished
S=-2 two-
baryon systems
23 November, November, 2018
On the nature of the X(3872)

14. No charge partners of the X(3872) [diquark-antidiquark]
DD*
DD* – J/ 
JPC(I)=1++(0)
T. Fernández-Caramés et al., Phys. Rev. Lett. 103, (2009)
X(3872)
No charge partners of the X(3872) [diquark-antidiquark]
JP=1+ and I=1, coupled to J/   Repulsive
23 November, November, 2018
On the nature of the X(3872)

15. Hadronic description DD*
C.E. Thomas, F.E. Close, Phys. Rev. D 78, (2009)
DD*
Hadronic description
Binding of the PV system is highly dependent on the parameters of a one-pion-exchange-like potential.
It is more difficult to bind exotic systems, what is clearly against the quark model expectations!
23 November, November, 2018
On the nature of the X(3872)

16. D D – c  D* D* – J/   23 November, 201823 November, 2018
On the nature of the X(3872)

17. I=1
- S. H. Lee et al. NPA815, 29 (2009). (QCD sum rules) Prefers the existence of a (IG)JP = (1-)1- D1D molecular state compatible with a bump around 4250 MeV. The Z(4050) being a threshold effect.
- Z.-G. Wang. EPJC59, 675 (2009). (QCD sum rules using different interpolating fields) prefers a scalar JP=0+ tetraquark for the Z(4250), obtaining a mass of 5.12 GeV for the 1-.
- G.J. Ding. PRD79, (2009). (meson exchange model). Concludes the Z(4250) cannot be a meson-meson molecule. It finds a strong attraction in the JP=0+ D*D* compatible with the Z(4050).
- X. Liu et al. EPJC61, 411 (2009). (meson exchange model). The interpretation of the Z(4050) as a D*D* molecule is not favored.
- D. Ebert et al. EJPC58, 399 (2008). (relativistic quark-antidiquark). Does not find a tetraquark candidate for the Z(4050) and suggest that the Z(4250) could be a JP = 0- or 1- hidden charm tetraquark.
23 November, November, 2018
On the nature of the X(3872)

18. JPC(I)=2++(1) D-wave S-wave Z+(4050)
T. Fernández-Caramés et al., Phys. Rev. D. 82, (2010)
D-wave
S-wave
Only one possible charged state has been found below 4.3 GeV
23 November, November, 2018
On the nature of the X(3872)

19. ! ! System JPC(I) DD 0++(0) DD* 1++(0) D*D* 2++(0) 2++(1)
R.Mizuk et al., Phys. Rev. D78, (2008)
T.F. Caramés, A.V., J.V., Phys. Rev. D82, (2010) !
Attractive channels for the two D-meson system
PRL67, 556 (1991) N.A. Törnqvist. PV and VV two-meson systems are the most natural candidates to be bound, in spite of the different working framework.
Y(3940)  T. Branz et al. PRD 80, (2009). D*D* JPC(I)=0++ (0)[2++(0)]. Effective lagrangians.
[Y(3940) J/ ]> 1 MeV.
Y(4140)  T. Branz et al. PRD 80, (2009). D*sD*s JPC(I)=0++ (0)[2++(0)]. Effective lagrangians.
[Y(4140) J/ ]> 1 MeV.
R.M. Albuquerque et al. PLB 678, 186 (2009). D*sD*s JPC(I)=0++ (0). QCD sum rules.
G.-J. Ding. EPJC 64, 297 (2009). D*sD*s JPC(I)=0++ (0). One-boson exchange model.
Y(3940), Z(3940), X(4160)  R. Molina et al. PRD 80, (2009). D*D* D*sD*s JPC(I)=0++ (0), 2++(0). Dynamically generated resonances.
23 November, November, 2018
On the nature of the X(3872)

20. Summary
The study of four-quark bound states must be based on exact solutions (approximate methods should be taken with care).
Within the minimal hypothesis scenario we have considered it is hard to conclude the existence of compact four-quark structures in systems with two different physical thresholds (QQnn) unless additional hypothesis are considered (diquarks, many.body terms, etc…)
Slightly bound (meson-meson molecules) four-quark states seem to be present in the heavy meson spectra.
PV: 1++ (0) are the candidate quantum numbers to lodge meson-meson molecules for systems made of non-identical mesons [X(3872)].
PP: 0++ (0) would the only candidate to lodge a broad meson-meson molecule for systems made of identical pseudoscalar mesons.
VV: 0++ (0) and 2++ (0,1) should show meson-meson molecules for systems made of identical vector mesons [Y(3940),Y(4140)].
23 November, November, 2018
On the nature of the X(3872) – –

21. 23 November, November, 2018
On the nature of the X(3872)

22. Backup
23 November, November, 2018
On the nature of the X(3872)

23. Beyond two-body interactions.
(II)
more stable than
becomes unstable for
23 November, November, 2018
On the nature of the X(3872)

24. 23 November, November, 2018
On the nature of the X(3872)

25. qqq+MB Pure qqq 23 November, 201823 November, 2018
P. Gonzalez et al.
qqq+MB
Pure qqq
23 November, November, 2018
On the nature of the X(3872)

26. 23 November, November, 2018
On the nature of the X(3872)

27. Charm mesons and charmonium
Light mesons
Strange meson
Charm mesons and charmonium
JV et al, J. Phys. G. 31, 481
Bottom mesons and bottomonium
Quarkonia properties at high temperatures
JV et al, Eur. Phys, Jour. A40, 89
23 November, November, 2018
On the nature of the X(3872)

28. Light baryons Heavy baryons Doubly Heavy baryons Strange baryons
JV et al., Phys. Rev. C72,
A. Valcarce et al., Eur. Phy. J. A37, 217
Doubly Heavy
baryons
Strange baryons
JV et al., Phys. Rev. D70,
23 November, November, 2018
On the nature of the X(3872)

29. Ligth scalars cccc states Dsj Bsj 23 November, 201823 November, 2018
N. Barnea et al., Phys. Rev. D73,
JV et al., Phys. Rev. D72,
Dsj
Bsj
JV et al., Phys. Rev. D73,
JV et al., Phys. Rev. D77,
23 November, November, 2018
On the nature of the X(3872)

30. where for practical purposes we have used the convention
23 November, November, 2018
On the nature of the X(3872)

31. Behaviour of the radius
← Molecular state
← Unbound state
← Bound state
Unbound state?
23 November, November, 2018
On the nature of the X(3872)

32. J. Vijande et al., Phys. Rev. D76, 094022 (2007)
cncn (I=0). BCN model
4q Energy
Theoretical threshold
J. Vijande et al., Phys. Rev. D76, (2007)
23 November, November, 2018
On the nature of the X(3872)

33. 23 November, November, 2018
On the nature of the X(3872)

34. Uncoupled Threshold ≥ Coupled Threshold
Thresholds
Uncoupled two-meson Threshold: Impose L, S, J, I, P, C (when defined) conservation, and the spin-statistic theorem (when identical particles are considered).
Coupled two-meson Threshold: Impose J, I, P, C (when defined) conservation, and the spin-statistic theorem (when identical particles are considered).
Uncoupled Threshold ≥ Coupled Threshold
23 November, November, 2018
On the nature of the X(3872)

35. cncn. CQC Model 4q Energy Uncoupled threshold Coupled threshold
23 November, November, 2018
On the nature of the X(3872)

36. Weak and electromagnetic decays
Weak decay
23 November, November, 2018
On the nature of the X(3872)

37. Candidates for observation (QQnn).
Decay modes.
Electromagnetic:
E4q > M(D)+M(D)
Weak:
E4q < M(D)+M(D)
Charm Sector: ccnn
1: JP=1+: CQC: ΔE= –76, ΔR= Compact. Weak decay
I=0 BCN: ΔE= –7, ΔR~1 – 2. Molecular. γ decay
Bottom Sector: bbnn
1: JP=1+: CQC: ΔE= –214, ΔR= Compact. Weak decay
I=0 BCN: ΔE= –144, ΔR= Compact. Weak decay
2: JP=0+: CQC: ΔE= –149, ΔR= Compact. γ decay
I=0 BCN: ΔE= –52, ΔR= Compact. γ decay
3: JP=3 – : CQC: ΔE= –140, ΔR= Compact. γ decay
I= BCN: ΔE= –119, ΔR= Compact. γ decay
4: JP=1 – : CQC: ΔE= –11, ΔR ~1 – 2. Molecular. Weak decay
I=0
23 November, November, 2018
On the nature of the X(3872)

38. Capabilities of the HH and VM methods.– –
L=0 S=1 I=0 ccnn – –
L=0 ccnn states
HOD*
SVA* HH
3931.0
3904.7
3899.2
(S,I)
VMCT*
HH (ℓi=0) HH
(0,1)
4155
4154
3911
(1,0)
3927
3926
3860
(1,1)
4176
4175
3975
(2,1)
4195
4193
4031
SVA*: Stochastic variational approach (BCN).
HOD*: Diagonalization in a harmonic oscillator basis up to N=8 (BCN). HH VM E
RMS
3860.6
0.367
3861.4
0.363
VMCT*: Variational calculation using gaussian trial wave functions with only quadratic terms in the Jacobi coordinates (CQC). .
23 November, November, 2018
On the nature of the X(3872)

39. Experimental Thresholds
cncn (I=0). BCN Model
Theorerical Thresholds
Experimental Thresholds 5!
23 November, November, 2018
On the nature of the X(3872)

40. Many-body forces in the hadron spectra( ) a y x L V r
MIN MB ij j i B 46 . 5 2 3 4 21 1 8 16 23 14 24 13 34 12 + = ø ö ç è æ - å
< l a a x x x a
23 November, November, 2018
On the nature of the X(3872)

41. Capability of the HH method (II)
L=0 ccnn states (MeV) –
L=0 cncn states (MeV) –
(S,I)
VMCT*
HH (ℓi=0) HH
(0,1)
4155
4154
3911
(1,0)
3927
3926
3860
(1,1)
4176
4175
3975
(2,1)
4195
4193
4031 JP
HOD* (N=8)
HH (K=8) HH 0+
3409
3380
3249 1+
3468
3436
3319
HOD*: Diagonalization in a harmonic oscillator basis up to N=8 (BCN).
VMCT*: Variational calculation using gaussian trial wave functions with only quadratic terms in the Jacobi coordinates (CQC). .
23 November, November, 2018
On the nature of the X(3872)

42. Color-Spin basis with well defined C-parity
The difficult part is to construct a symmetrized color-spin basis for the N-body system with well defined C-parity.
Spin part → One make use of the SU(2) Clebsh-gordan coefficients.
Color part → One could be temped to use the SU(3) Clebsh-gordan coefficients, however this is not feasible.
To construct the color part we have used a method based on an algorithm by Novoselsky, Katriel and Gilmore (J. Math. Phys. 29, 1368), obtaining states with well defined permutational symmetry, spin and color proyection. 
Evaluating the quadratic casimir SU(3) operator one can determine the specific representation of each state, picking only those belonging to the SU(3) singlet.
Evaluating the charge conjugation operator one can choose only those states with well defined C-parity, c = ±1.
23 November, November, 2018
On the nature of the X(3872)