The Direct Coupling of Light Quarks to Heavy Di-quarks

Abstract

In the limit $m_Q>m_Q{v}_{\mathrm{rel}}>m_Q{v}_{\mathrm{rel}}^2\gg {\Lambda }_{QCD}$ 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 $\left(S_v^{†}{S}_v\right)\left(\bar{q}{\gamma }^{\mu }{v}_{\mu }q\right)$ 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 $\mathcal{O}(1/({\alpha }_s(m_Q{v}_{\mathrm{rel}})m_Q^2))$. 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.

Comments: 11 pages, 2 figures