Color gauge invariant theory of diquark interactions

In the present work, we construct a color gauge invariant theory of diquark interactions. With the transformation rule of the diquark fields and the definition of the covariant derivatives under color SU(3) symmetry, we construct the gauge invariant Lagrangians describing the vertices of $S-S-G$, $A-A-G$, $S_Q-S_Q-G$, $A_Q-A_Q-G$ and $S_Q-A_Q-G$, where $S_{(Q)}$ is the light (heavy) scalar diquark field, $A_{(Q)}$ is the light (heavy) axial vector diquark field, and $G$ is the gluon field. And then we derive the one-gluon-exchange effective potentials for diquark-antidiquark interactions using the obtained Lagrangians. By comparing these potentials with those in Godfrey-Isgur quark model, the coupling constants of the Lagrangians are determined. We find that the potentials are mainly made of Coulomb, contact and tensor terms. The potential for the process $S\bar{A}\to A\bar{S}$ is 0, since $\mathcal{L}_{SAG}=0$. And the tensor terms proportional to $g^2$ are negligible.