On the smooth cross-over transition from the Delta-string to the Y-string three-quark potential

TL;DR

This paper critically examines the claim of a smooth transition between Delta-string and Y-string three-quark potentials, demonstrating that their differing symmetries prevent a true cross-over except at zero hyper-radius.

Contribution

The study shows that the Delta-string and Y-string potentials have different symmetries and belong to different universality classes, challenging previous assertions of a smooth transition.

Findings

Different symmetries lead to different constants of motion.

No smooth cross-over exists except at zero hyper-radius.

Y-string and Delta-string potentials are in different universality classes.

Abstract

We comment on the assertion made by Caselle et al., J. Stat. Mech. 0603, P008 (2006), that the confining (string) potential for three quarks "makes a smooth cross-over transition from the Delta-string to the Y-string configuration at interquark distances of around 0.8 fm". We study the functional dependence of the three-quark confining potentials due to a Y-string, and the Delta-string and show that they have different symmetries, which lead to different constants of the motion (i.e. they belong to different "universality classes" in the parlance of the theory of phase transitions). This means that there is no "smooth cross-over" between the two, when their string tensions are identical, except at the vanishing hyper-radius. We also comment on a certain two-body potential approximation to the Y-string potential.