1. Mitja Rosina THE BINDING ENERGY OF THE Ξcc++ BARYON
AND THE DD* DIMESON
Mitja Rosina
Faculty of Mathematics and Physics, University of Ljubljana
and Institute Jožef Stefan, Ljubljana, Slovenia
Talk presented at the Mini-Workshop BLED, June, 2018
DOUBLE-CHARM BARYONS AND DIMESONS
OUR ESTIMATE: m( Ξcc++ ) ≥ or MeV
LHCb: m( Ξcc++ ) = 3621 MeV
SELEX: m( Ξcc+ ) = 3519 MeV

2. DECAY CHANNELS: c → s LHCb: m( Ξcc++ ) = 3621 MeV
Ξcc++ → Λc+ + K- + π+ + π+ , Λc+ → p + K- + π+
ccu → cud + su + ud + ud
SELEX: m( Ξcc+ ) = 3519 MeV
Ξcc+ → Λc+ + K- + π+
Ξcc+ → D+ + p + K-

5. or PHENOMENOLOGICAL ESTIMATE OF THE Ξcc++ MASS u u u b c c c
like D0 meson
( ) MeV
like B+ meson
( ) MeV
m (c) = MeV m (b) = 5259 MeV
m (cc) = MeV
m (ccu) = m(D0) - m(c) + m(cc) m (ccu) = m(B+) - m(b) +m(cc)
= =3532 MeV = =3557 MeV

6. Vcc = 1 2 Vcc The „Vqq = Vqq ‘‘ rule for diquark meson12
Vcc = Vcc
[A]: (color.color)=4/3 for cc
=2/3 for cc
[B]: Flux tube model c c c c c

7. 𝑝2 2(𝑐/2) +𝑉𝑐𝑐 𝜓 = Ecc 𝜓 𝑝2 2(𝑐/2) + 𝑉𝑐𝑐 𝜓 = 1 2
The „Vqq = Vqq ‘‘ rule for diquark meson 12
𝑝2 2(𝑐/2) +𝑉𝑐𝑐 𝜓 = Ecc 𝜓
𝑝2 2(𝑐/2) + 𝑉𝑐𝑐 𝜓 = 1 2
𝑝𝟐 2(𝑐/4) +𝑉𝑐𝑐 𝜓 = Ecc 𝜓
Ecc = F(c/2)
Ecc = F(c/4)
notation: particle symbols mean their masses

8. The finite size of the diquark and the extra Coulomb repulsion would raise the mass a little.
The estimate (≥ or MeV) is then not too far from the experimental value MeV.
It is considerably above the SELEX value 3519 MeV, but it does not exclude it.

10. BINDING ENERGY OF THE DD* „MOLECULE“
The nonrelativistic calculation of Janc & Rosina (2004)
Used the one-gluon exchange potential (including the chromomagnetic term) + the linear confining potential.
The model parameters fitted all relevant mesons and baryons.
Janc & Rosina : (DD*) – (D + D*) = MeV (Bhaduri )
- 2.7 MeV (Grenoble AL1)
Important questions:
1. How much is binding for other interactions?
2. Can the pion cloud between the u and d antiquarks
(the DD*π or DD*ππ configuration) increase binding ?
3. How much does relativity change binding?

11. MOTIVATION
Effective quark-quark interaction:
Is Vuu = Vcu = Vcc = Vcc = Vbu = Vbb = Vbb
(apart from the mass-dependent spin-spin term) ?
Interesting 4-body problem,
to understand binding and to forsee the production
and detection mechanisms.
CHALLENGE:
New chances to measure LHC and DD*=(cu+cd) at the upgraded KEK (Tsukuba, Japan).

12. COMPARISON BETWEEN DIMESON AND HYDROGEN MOLECULE POTENTIALV
rcc u cc c c c
D + D* d

13. PHENOMENOLOGICAL ESTIMATE OF DIMESON BINDINGu u b u bb b b d b d d
Dimeson B+B*
( =10604)MeV
like Λb antibaryon ( )MeV
Tetraquark u u c u c c cc d c d d
like Λ c antibaryon ( )MeV
Dimeson D+D*
( =3875)MeV
Tetraquark

14. Is the D+D* dimeson bound?
In the restricted 4-body space assuming "cc" in a bound diquark state + general wavefunction of u and d, the energy is above the D+D* threshold.
In the restricted "molecular" 4-body space assuming the two c quarks far apart + general wavefunction of ubar and dbar (as assumed by several authors), the energy is also above the D+D* threshold.
Only combining both spaces (we took a rich 4-body space) brings the energy below the threshold.

15. Gaussian basis with Jacobi coordinates

16. The nonrelativistic calculation of Janc & Rosina (2004)
Used the one-gluon exchange potential (including the chromomagnetic term) + the linear confining potential.
The model parameters fitted all relevant mesons and baryons.
A rich 4-body space was used (an s-state Gaussian expansion at optimized distances, with 3 types of Jacobi coordinates in order to mimic also the p-states).
(DD*) – (D + D*) = MeV (Bhaduri )
- 2.7 MeV (Grenoble AL1)
Can the pion cloud between the u and d antiquarks increase binding, in analogy with the deuteron ?
At a distance of 0.5 fm the two-pion-exchange interaction
between the D and D* mesons might bring additional 50 MeV??

17. A note on analogy between DD* and DD*
We failed to calculate the energy of X(3872) using the same method and interaction as for DD*, [Bled Proceedings 2005]. The reason is that a perfect variational calculation in a rather complete 4-body space finds the absolute minimum of energy which correspnds to J/psi+eta rather than DD* A demanding coupled channel calculation would be needed for a reliable result, and we have posponed it.