Few-body quark dynamics for doubly heavy baryons and tetraquarks

We discuss the adequate treatment of the three- and four-body dynamics for the quark model picture of double-charm baryons and tetraquarks. We stress that the variational and Born-Oppenheimer approximations give energies very close to the exact ones, while the diquark approximation might be somewhat misleading. The Hall-Post inequalities also provide very useful lower bounds that exclude the possibility of stable tetraquarks for some mass ratios and some color wave functions.

Figure 1

Comparison of different approximations for a tetraquark bound by a linear potential with frozen color wave function, or with color mixing. Dotted blue line, pure $6\overline{6}$; dashed blue line, pure $\overline{3}3$; solid blue line, with color mixing; red line, threshold. The units are such that $m=1$ and $v_{ij}=r_{ij}$.

Figure 2

Same as Fig. 1 for a Coulomb-plus-linear interaction $v\left(r\right)=-0.4/r+0.2r$, where $r$ is in ${\mathrm{GeV}}^{-1}$. The energy $E$ is in GeV.

Figure 3

Comparison of the exact energy (solid blue line) and diquark approximation (red dotted line) for a baryon $\left(QQq\right)$ with masses $M$ and $m=1$, and a purely linear interaction, as a function of the mass ratio. The units are such that $m=1$ and the potential is $\sum r_{ij}/2$.

Figure 4

Comparison of the variational upper bound (green curve) and Hall-Post lower bound (dotted blue curve), hardly distinguishable from the variational estimate at this scale, for the tetraquark Hamiltonian (1) with a linear interaction. Also shown is the naive diquark-antidiquark approximation (dashed violet curve).

Figure 5

Same as Fig. 4 with a potential $r_{ij}^{0.1}$ instead $r_{ij}$.

Figure 6

Same as Fig. 4 with a potential $-r_{ij}^{-1}$ instead $r_{ij}$. The threshold is fixed at $-2$.

Figure 7

Comparison of the light quark energies for $QQ\overline{q}\overline{q}$ (solid red line) and $QQq$ (dotted blue line) as a function of the $QQ$ separation $x$. The second curve is shifted by the difference of energies $Qqq-Q\overline{q}$. The units are such $m=1,M=5$, and $v_{ij}=r_{ij}$.

Figure 8

Effect of color mixing on the binding of $QQ\overline{u}\overline{d}$, within the AL1 model. The tetraquark energy is calculated with only the color $\overline{3}3$ configurations (blue curve) and with the $6\overline{6}$ components (green curve).

Figure 9

Effect of the spin-spin interaction of the binding of $QQ\overline{u}\overline{d}$, within the AL1 model. The tetraquark energy is calculated with (green line) and without (blue line) the chromomagnetic term.