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, 17-23 June, 2018 DOUBLE-CHARM BARYONS AND DIMESONS OUR ESTIMATE: m( Ξcc++ ) ≥ 3532 or 3557 MeV LHCb: m( Ξcc++ ) = 3621 MeV SELEX: m( Ξcc+ ) = 3519 MeV

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-

or PHENOMENOLOGICAL ESTIMATE OF THE Ξcc++ MASS u u u b c c c like D0 meson (1865 3532) MeV like B+ meson (5279 3557) MeV m (c) = 1870 MeV m (b) = 5259 MeV m (cc) = 3537 MeV m (ccu) = m(D0) - m(c) + m(cc) m (ccu) = m(B+) - m(b) +m(cc) =1865-1870+3537 =3532 MeV =5279-5259+3537=3557 MeV

Vcc = 1 2 Vcc The „Vqq = Vqq ‘‘ rule for diquark meson 12 Vcc = 1 2 Vcc [A]: (color.color)=4/3 for cc =2/3 for cc [B]: Flux tube model c   c c c c

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 = 1 2 F(c/4) notation: particle symbols mean their masses

The finite size of the diquark and the extra Coulomb repulsion would raise the mass a little. The estimate (≥ 3532 or 3557 MeV) is then not too far from the experimental value 3621 MeV. It is considerably above the SELEX value 3519 MeV, but it does not exclude it.

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*) = - 0.6 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?

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 BB*=(bu+bd) @ LHC and DD*=(cu+cd) at the upgraded Belle-2 @ KEK (Tsukuba, Japan).

COMPARISON BETWEEN DIMESON AND HYDROGEN MOLECULE POTENTIAL V rcc u cc c c c D + D* d

PHENOMENOLOGICAL ESTIMATE OF DIMESON BINDING u u b u bb b b d b d d Dimeson B+B* (5279+5325=10604)MeV like Λb antibaryon (5620 10476)MeV Tetraquark u u c u c c cc d c d d like Λ c antibaryon (2286 3967)MeV Dimeson D+D* (1867+2008=3875)MeV Tetraquark

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.

Gaussian basis with Jacobi coordinates

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*) = - 0.6 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??

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.