Smooth crossover transition from theΔ-string to theY-string three-quark potential

Abstract

We comment on the assertion made by Caselle et al. [M. Caselle, G. Delfino, P. Grinza, O. Jahn, and N. Magnoli, J. Stat. Mech. (2006) P008.] that the confining (string) potential for three quarks ``makes a smooth crossover transition from the $\ensuremath{\Delta}$-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 $\ensuremath{\Delta}$ 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 crossover'' 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.