Some Aspects of Three-Quark Potentials (Part II)

TL;DR

This work extends a holographic, soft-wall AdS/QCD approach to compute the three-quark potential for triply heavy systems in isosceles and collinear geometries, deriving detailed UV and IR behaviors and illustrating a universal string tension. It analyzes three basic string configurations in the isosceles case, showing how the potential interpolates between a $\Delta$-like regime at short distances and a true three-body $Y$-law at large separations, with a universal leading slope $\sigma$ and geometry-dependent constants. The IR analysis reveals two distinct regimes: for largest angles $\leq 2\pi/3$, the $Y$-law dominates with calculable leading corrections; for obtuse triangles with angle $> 2\pi/3$, two strings dominate and the constant term encodes the geometry, hinting at a permutation-symmetry breaking to a heavy object in the two-index antisymmetric representation. The results provide connections to lattice QCD, quantify diquark approximations in the collinear case, and illuminate how baryonic flux-tube configurations fare in nonperturbative QCD, with potential implications for heavy baryon phenomenology.

Abstract

We continue our investigation of the effective string model for the triply heavy quark system, mimicking that in pure $SU(3)$ gauge theory. We present analytical and numerical studies of the three-quark potential for isosceles and collinear geometries. In the general case, we derive the asymptotic expression of the potential in the infrared limit. Here we also demonstrate the universality of the string tension and interpret the transition between two distinct regimes, occurring when one of the triangle's angles formed by the quarks is equal to $\frac{2}{3}π$, as a breaking of permutational symmetry. This symmetry breaking implies the emergence of a heavy quark dressed by gluons, transforming in the two-index antisymmetric representation. Additionally, we discuss various aspects of the $Y$- and $Δ$-laws, diquarks, and connections to lattice QCD.

Figures (24)

• Figure 1: A typical baryonic configuration in five dimensions. The heavy quarks are denoted by $Q_i$, the baryon vertex by $V$, its projection onto the boundary by $Y$, and the tangent angles of the strings by $\alpha_i$ ($i=1,\dots,3$). The arrows indicate the forces acting on the vertex, which is located in the bulk at $r=r_v$.
• Figure 2: An isosceles triangle with leg length $\ell$ and apex angle $\theta$. Here $\beta=\angle{YQ_1Q_2}$.
• Figure 3: Basic string configurations associated to the different signs of the tangent angles. In (I) $\alpha_1\geq 0$ and $\alpha_3\leq 0$. In (II) all the angles are non-positive. In (III) $\alpha_1\leq 0$ and $\alpha_3\geq 0$.
• Figure 4: $E_{\text{\tiny 3Q}}$ vs $l$ for several values of $\theta$ not exceeding $\frac{2}{3}\pi$. The dotted lines represent the asymptotic behavior \ref{['largeELI']}. We set $c=623\,\text{MeV}$, here and in all subsequent Figures.
• Figure 5: $E_{\text{\tiny 3Q}}$ vs $l$ for several values of $\theta$ exceeding $\frac{2}{3}\pi$.