Wilson loops, $q \bar{q}$ and $3 q$ potentials, Bethe--Salpeter equation

TL;DR

This paper reviews the derivation of quark-antiquark and three-quark potentials using Wilson loops, introduces improvements, and derives a relativistic flux-tube Lagrangian and Bethe--Salpeter equation under common assumptions.

Contribution

It provides a refined derivation of quark potentials and extends the framework to include a relativistic flux-tube Lagrangian and a confining Bethe--Salpeter equation for spinless quarks.

Findings

Derived $q ar q$ and $3q$ potentials using Wilson loops.

Obtained a relativistic flux-tube Lagrangian.

Formulated a confining Bethe--Salpeter equation for spinless quarks.

Abstract

The derivation of the $ q \bar q $ and the $ 3q $ potential for two dynamical quarks in a Wilson--loop context is reviewed. Some improvements are introduced. Only the usual assumptions in the evaluation of the Wilson loop integrals and expansions in the quark velocities are required for the result. It is shown that under the same assumptions it is possible to obtain the relativistic flux--tube lagrangian and a $ q \bar{q}$ Bethe--Salpeter equation with a confining kernel for spinless quarks. IFUM 482/FT