Color gauge invariant theory of diquark interactions

TL;DR

This work extends $SU(3)_{c}$ color gauge invariance to the diquark sector by constructing color gauge invariant Lagrangians for diquark–gluon interactions, including the vertices $S-S-G$, $A-A-G$, $S_Q-S_Q-G$, $A_Q-A_Q-G$, and $S_Q-A_Q-G$, with $S$ (scalar) and $A$ (axial vector) diquarks. Using these Lagrangians, it derives one-gluon-exchange potentials between diquarks and antidiquarks, which decompose into Coulomb, contact, and tensor terms, and notes that the $S\bar{A}\to A\bar{S}$ channel vanishes due to $\mathcal{L}_{SAG}=0$. Couplings $g$, $d_{1Q}$, $d_2$, and $d_{2Q}$ are fixed by matching to the Godfrey-Isgur quark model, obtaining $g\approx 2.7$, $d_{1Q}\approx 3.1\ \mathrm{GeV}^{-1}$, $d_2\approx 10.5$, and $d_{2Q}\approx 13.3\ (34.1)$ for charmed (bottom) diquarks. The results show that tensor terms are generally suppressed, and provide quantitative inputs for tetraquark studies within a gauge-invariant diquark framework.

Abstract

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.

Figures (1)

• Figure 1: The one-gluon-exchange interactions between diquark and antidiquark at the diquark level. $S_{(Q)}$ denotes the light (heavy) scalar diquark, and $A_{(Q)}$ the light (heavy) axial vector diquark. $S_{(Q)}$ stands for $S$ or $S_Q$, and $A_{(Q)}$ for $A$ or $A_Q$.