A consistent derivation of the quark--antiquark and three quark potentials in a Wilson loop context

TL;DR

This paper presents a new, explicit derivation of quark-antiquark and three-quark potentials within the Wilson loop framework, clarifying assumptions and extending previous results to include spin-dependent and independent terms accurately.

Contribution

It provides a consistent derivation of both quark-antiquark and three-quark potentials, correcting previous assumptions and extending the potential forms to include spin effects up to order 1/m^2.

Findings

Reproduces known semirelativistic potential for qar{q} with clarified ordering.

Derives spin-dependent three-quark potential matching literature but with correct assumptions.

Obtains the accurate form of the spin-independent potential up to 1/m^2 order.

Abstract

In this paper we give a new derivation of the quark-antiquark potential in the Wilson loop context. This makes more explicit the approximations involved and enables an immediate extension to the three-quark case. In the $q\overline{q}$ case we find the same semirelativistic potential obtained in preceding papers but for a question of ordering. In the $3q$ case we find a spin dependent potential identical to that already derived in the literature from the ad hoc and non correct assumption of scalar confinement. Furthermore we obtain the correct form of the spin independent potential up to the $1/m^2$ order.