Diquark size effects in the quark-diquark approximation for baryons

TL;DR

This work benchmarks the quark-diquark (QD) approximation against a full three-body baryon model using the same semi-relativistic, QCD-inspired potential. It introduces three QD implementations—point-like diquark, density-based convolution with $|\psi_D|^2$, and a density-operator convolution—showing that accurate baryon masses do not require a compact diquark. By deriving and applying analytic convolution formulas within the Lagrange-mesh framework, the study demonstrates that the density-based convolution $V_{Dq2}$ best reproduces three-body results across several $n$ and $b$-quark containing baryons, while characteristic distances reveal limitations in translating three-body structure to the QD picture. The findings imply that careful incorporation of diquark size into the potential, rather than assuming a point-like cluster, is crucial for reliable spectroscopy; they also point to directions for extending the approach to more complex multiquark systems. The work thus clarifies the domain of validity of the quark-diquark approximation and informs future model-building in hadron spectroscopy.

Abstract

Baryons can be described within several theoretical frameworks. Among them, the constituent approach is widely used. In this context, we aim to evaluate the accuracy of a particular model of baryons: the quark-diquark approximation. It consists in separating the three-body system into two subsequent two-body ones: a pair of two quarks, the diquark, and a second system consisting of the diquark and the third quark. This approximation is widely used, but its accuracy is rarely evaluated. The goal of this work is to perform this evaluation by comparing the quark-diquark model with a three-body model, both using the same semi-relativistic interaction. The baryon masses and some characteristic distances are computed and analysed within both approaches. Additionally, an original procedure to establish the quark-diquark potential will be presented with the aim to increase the precision of this approximation. It is shown that a diquark must not necessarily be compact to obtain good baryon masses.

Figures (6)

• Figure 1: Illustration of formula \ref{['eq: conv. int. 2']}.
• Figure 2: Comparison of the three quark-diquark potentials for potential \ref{['eq: Vqqbar test']} and wave function \ref{['eq: OH wf']}. The solid grey curve represents the unconvoluted potential profile. The blue dash-dot curve corresponds to the first convoluted potential $V_{Dq1}$, while the red dashed curve illustrates the second convoluted potential $V_{Dq2}$. All quantities are given in arbitrary units.
• Figure 3: Structures of the ground (left) and $L=8$ (right) states of the $bbb$ baryon obtained by computing the mean values of $\rho$ and $\lambda$. Both chosen states have a total spin $S=3/2$. The ratio defined in \ref{['eq: eta']} takes the values $\eta=1.152$ and $\eta=1.105$ for these states, respectively.
• Figure 4: Structures of the ground $S=3/2$ (left) and $S=1/2$ (right) states of the $nnn$ baryon obtained by computing the mean values of $\rho$ and $\lambda$. The ratio defined in \ref{['eq: eta']} takes the values $\eta=1.153$ and $\eta=1.151$ for these states, respectively.
• Figure 5: Structures of the ground (left) and $L=8$ (right) states of the $bbn$ baryon obtained by computing the mean values of $\rho$ and $\lambda$. Both chosen states have a total spin $S=1/2$, and a total isospin $I=1/2$. The ratio defined in \ref{['eq: eta']} takes the values $\eta=0.669$ and $\eta=2.056$ for these states, respectively.