The Direct Coupling of Light Quarks to Heavy Di-quarks

In the limit $m_Q >m_Q v_{\rm rel} > m_Q v_{\rm rel}^2 \gg \Lambda_{QCD}$ hadronic states with two heavy quarks $Q$ should be describable by a version of HQET where the heavy quark is replaced by a di-quark degree of freedom. In this limit the di-quark is a small (compared with $1/\Lambda_{QCD}$) color anti-triplet, bound primarily by a color Coulomb potential. The excited Coulombic states and color six states are much heavier than the color anti-triplet ground state. The low lying spectrum of hadrons containing two heavy quarks is then determined by the coupling of the light quarks and gluons with a momentum of order $\Lambda_{QCD}$ to this ground state di-quark. In this short paper we calculate the coefficient of leading local operator $\left( S_v^{\dagger} S_v\right) \left({\bar q} {\gamma^{\mu} v_{\mu} } q\right)$ that couples this color-triplet di-quark field $S_v$ (with four-velocity $v$) directly to the light quarks $q$ in the low energy effective theory. It is ${\cal O}(1/( \alpha_s(m_Q v_{\rm rel}) m_Q^2))$. While our work is mostly of pedagogical value we make an estimate of the contribution of this operator to the masses of $\Xi_{bbq}$ baryon and $T_{QQ{\bar q} {\bar q}}$ tetraquark using the non-relativistic constituent quark model.