Iterative method to compute the Fermat points and Fermat distances of multiquarks

TL;DR

This paper introduces an iterative method for efficiently computing Fermat points and distances in multiquark systems, extending geometric solutions to complex multiquark configurations.

Contribution

It presents a novel iterative approach for calculating Fermat points in multiquark systems, generalizing geometric solutions to complex configurations.

Findings

The method converges rapidly to accurate Fermat points.

It simplifies the computation of multiquark potential distances.

Provides a review of geometric Fermat point solutions.

Abstract

The multiquark confining potential is proportional to the total distance of the fundamental strings linking the quarks and antiquarks. We address the computation of the total string distance an of the Fermat points where the different strings meet. For a meson (quark-antiquark system) the distance is trivially the quark-antiquark distance. For a baryon (three quark system) the problem was solved geometrically from the onset, by Fermat and by Torricelli. The geometrical solution can be determined just with a rule and a compass, but translation of the geometrical solution to an analytical expression is not as trivial. For tetraquarks, pentaquarks, hexaquarks, etc, the geometrical solution is much more complicated. Here we provide an iterative method, converging fast to the correct Fermat points and the total distances, relevant for the multiquark potentials. We also review briefly the geometrical methods leading to the Fermat points and to the total distances.