BB intermeson potentials in the quark model

In this paper we derive quark model results for scattering amplitudes and equivalent low energy potentials for heavy meson pairs, in which each meson contains a heavy quark. This “BB” system is an attractive theoretical laboratory for the study of the nuclear force between color singlets; the hadronic system is relatively simple, and there are lattice gauge theory (LGT) results for $V_{\mathrm{BB}}\mathrm{}\left(r\right)\mathrm{}$ which may be compared to phenomenological models. We find that the quark model potential (after lattice smearing) has qualitative similarities to the LGT potential in the two $B^{*}{B}^{*}$ channels in which direct comparison is possible, although there is evidence of a difference in length scales. The quark model prediction of equal magnitude but opposite sign for $I=0\mathrm{}$ and $I=1\mathrm{}$ potentials also appears similar to LGT results at intermediate r. There may however be a discrepancy between the LGT and quark model $I=1\mathrm{}$ $\mathrm{BB}$ potentials. A numerical study of the two-meson Schrödinger equations in the $\left(b\overline{q}\right)\left(b\overline{q}\right)$ and $\left(c\overline{q}\right)\left(c\overline{q}\right)$ sectors with the quark model potentials finds a single BB “molecule,” in the $I=0\mathrm{}$ ${\mathrm{BB}}^{*}$ sector. Binding in other channels might occur if the quark model forces are augmented by pion exchange.