Bled Workshops in Physics Vol. 5, No. 1
A
Proceedings of the Mini-Workshop
Quark Dynamics (p. 70)
Bled, Slovenia, July 12-19, 2004

Molecular binding of Tcc = DD* tetraquark *
Damijan Janca and Mitja Rosinaa,b bJ. Stefan Institute, 1000 Ljubljana, Slovenia
Q Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
Abstract. We present the results of detailed calculations with Bhaduri and AL1 potential for the Tcc = DD * tetraquark. We show that it has a molecular structure, which can transform, under the influence of an additional three-body force, into a Ab-like system where the role of the b quark is played by the heavy cc diquark.
Nonrelativistic potential models have proven to be quite a successful tool for understanding the meson and baryon sector. It is challenging to extend them from one-hadron to two hadron systems, such as the double heavy tetraquarks. Probably the most intriguing tetraquark in this class is the Tcc = DD* tetraquark. The results obtained with different potential models are very contradictive, from unbound [1,2] to deeply bound states [3,4]. If one demands, however, that the model used in the calculations must reproduce accurately the meson as well as baryon sector, then we believe that the dependence of the results on the model should not be so strong. Actually, the results should only be sensitive to the details of the interaction, which are not of the great importance for the meson or baryon sector, such as for example the colour dependent three-body force.
We present the results obtained with two one-gluon-exchange potentials, the Bhaduri [5] and Grenoble AL1 [6] potential. For a long time it was supposed that Tcc is unbound with these two potentials, according to seemingly accurate calculations [2,7]. We expanded the tetraquark wavefunction in 140 Gaussians of optimized widths for three sets of Jacobi coordinates to obtain 0.1 MeV accuracy (Fig.1) and show [8,9], however, that with both, the Bhaduri and the Grenoble AL1 potentials, Tcc is bound below the DD* threshold by 0.6 and 2.7 MeV, respectively.
It is essential to use a large enough model space to accommodate the molecular structure, in contradistinction to Tbb which has an atomic structure similar to Ab. Both types of configurations are schematically illustrated in Fig.2. If the basis is too small the Tcc tetraquark without additional interactions remains unbound. This had happened in [10], where the same basis functions were used as here, but the final basis was spanned with only 40 functions, since so extremely weak binding was not expected. From Fig.1 we see that at least 80 basis function are needed to obtain the energy of the DD* system lower than the threshold.
* Talk delivered by D. Janc.
Tcc = DD* molecule 71
3910
0 20 40 60 80 100 120 140 n
Fig.1. Energy of the Tcc tetraquark with Bhaduri potential as a function of the number of the basis states for three different runs. The D + D* threshold is also shown. Since the initial parameters are chosen randomly, the convergence is similar as with the stochastic variational approach.
3909
3908
3907
3906
3905
In Fig.3a we plot the probability densities Pqq between heavy quarks in Tbb and Tcc as a function of the interquark distance tqq:
PQQ(r) = (^I§(r - TQQ)|^).
a)	b)
Fig. 2. Schematic illustration of the two light antiquarks (empty circles) and two heavy quarks (dashed circles) in a): atomic configuration as we can find it in the Tbb tetraquark and in b): molecular configuration characteristic for the Tcc tetraquark.
There are also other mechanisms to help binding: 3-body forces (which are more effective for 4 particles than for 3 particles - baryons) and pion exchange (pions are almost real when exchanged between D and D* mesons). The form of the three-body interaction which we introduced into the tetraquark is
Vq3bqq (n, rj, rk) =	AfA^Uo expH^ + rfk + r^/r2],
V^q(ri, rj.ru) = ldttbcAf Ab*A^U0 expH^ + ifk + r^ )/r2].
Here r^ is the distance between i-th and j-th (anti)quark, and similarly for rjk and rki. Aa are the Gell-Mann colour matrices and dabc are the SU(3) structure constants ({Aa, Ab} = 2dabcAc).
It should be noted that in the baryon sector such a colour structure is irrelevant since there is only one colour singlet state and thus the colour factor
72	D. Janc. M. Rosina
Fig.3. (a): Tbb shows atomic structure while Tcc is molecular, r = rbb or rcc; (b): The effect of three-body interaction on the structure of Tcc for 3 different strengths.
is just a constant which can be included into the strength of the potential. In tetraquarks the situation is different since there are two colour singlet states: 312334 and 612 634 (or 1i3124 and 813824 after recoupling). The three-body force operates differently on these two states [11,12] and one can anticipate that in the case of the weak binding it can produce large changes in the structure of the tetraquark. This cannot be otherwise produced simply by reparameterization of the two-body potential, so the weakly bound tetraquarks are a very important laboratory for studying the effect of such an interaction.
It is well known that the constituent quark models with the colour A ■ A structure give rise to the long-range van der Waals forces [13-15], which can have dramatic effect especially for weakly bound systems with the molecular structure, such as the Tcc tetraquark. This interaction appears due to the colour polarization of two mesons in the colour singlet state and is an artefact of the potential approach. It is not present in the full QCD where quark-anriquark pair creation from the confining filed energy would produce an exponential cut-off of this residual interaction. The radial dependence has in the case of the linear confining interaction the structure
V (r)v.d.Waais = O(rd-4 ) = O(r-3)
We now check the effect of this spurious interaction in the Tcc tetraquark. In Fig. 4 we present useful quantity, which we call effective potential density
vij (r) = MVj (rtj )5(r - ry )|^) = Vj (r)Pij (r).	(1)
In Fig. 4b one can see that this effect is indeed present at large separations (r > 2 fm) but is extremely small. Integrating this attractive tail of the potential, we obtain a contribute less than 100 keV to the binding of the system. On the other hand, more interesting feature of the effective potential shown in Fig. 4 is the repulsive force between quarks at the medium distance between quarks (1.5 fm> r > 2 fm). The maximal value of potential barrier is Vy (r ~ 1.5 fm) = Vij/Pij = 1 MeV. This then allows that also resonant states can appear in the model which are not possible in a simple potential well.
Tcc = DD* molecule 73
Fig. 4. Left: Potential densities vij between (anti)quarks as calculated from Eq. 1 for Bhaduri potential. Right: Enlarged section of the left-hand side figure, where van der Waals attraction and medium-range repulsion can be seen.
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